diff --git a/NEWS.md b/NEWS.md
index 8cdb6e7a..41eb7873 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,12 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.15.17] 04/10/2024
+
+### Changed
+
+* **Mildly breaking**: the number system parameter now corresponds to the coefficients standing in front of basis vectors in a linear combination instead of components of a vector. For example, `DefaultOrthonormalBasis() == DefaultOrthonormalBasis(ℝ)` of `DefaultManifold(3, field=ℂ)` now has 6 vectors, and `DefaultOrthonormalBasis(ℂ)` of the same manifold has 3 basis vectors.
+
 ## [0.15.16] 13/09/2024
 
 ### Changed
diff --git a/Project.toml b/Project.toml
index bf7b878d..ad144c1a 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "ManifoldsBase"
 uuid = "3362f125-f0bb-47a3-aa74-596ffd7ef2fb"
 authors = ["Seth Axen <seth.axen@gmail.com>", "Mateusz Baran <mateuszbaran89@gmail.com>", "Ronny Bergmann <manopt@ronnybergmann.net>", "Antoine Levitt <antoine.levitt@gmail.com>"]
-version = "0.15.16"
+version = "0.15.17"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/ext/ManifoldsBaseQuaternionsExt.jl b/ext/ManifoldsBaseQuaternionsExt.jl
index 742ec168..be45760d 100644
--- a/ext/ManifoldsBaseQuaternionsExt.jl
+++ b/ext/ManifoldsBaseQuaternionsExt.jl
@@ -2,13 +2,13 @@ module ManifoldsBaseQuaternionsExt
 
 if isdefined(Base, :get_extension)
     using ManifoldsBase
-    using ManifoldsBase: ℍ
+    using ManifoldsBase: ℍ, QuaternionNumbers
     using Quaternions
 else
     # imports need to be relative for Requires.jl-based workflows:
     # https://github.com/JuliaArrays/ArrayInterface.jl/pull/387
     using ..ManifoldsBase
-    using ..ManifoldsBase: ℍ
+    using ..ManifoldsBase: ℍ, QuaternionNumbers
     using ..Quaternions
 end
 
@@ -20,4 +20,12 @@ end
     return QuaternionF64
 end
 
+@inline function ManifoldsBase.coordinate_eltype(
+    ::AbstractManifold,
+    p,
+    𝔽::QuaternionNumbers,
+)
+    return quat(number_eltype(p))
+end
+
 end
diff --git a/src/DefaultManifold.jl b/src/DefaultManifold.jl
index cbaa0029..6bf2f41d 100644
--- a/src/DefaultManifold.jl
+++ b/src/DefaultManifold.jl
@@ -65,23 +65,22 @@ embed!(::DefaultManifold, Y, p, X) = copyto!(Y, X)
 
 exp!(::DefaultManifold, q, p, X) = (q .= p .+ X)
 
-function get_basis_orthonormal(::DefaultManifold, p, N::RealNumbers)
-    return CachedBasis(
-        DefaultOrthonormalBasis(N),
-        [_euclidean_basis_vector(p, i) for i in eachindex(p)],
-    )
-end
-function get_basis_orthogonal(::DefaultManifold, p, N::RealNumbers)
-    return CachedBasis(
-        DefaultOrthogonalBasis(N),
-        [_euclidean_basis_vector(p, i) for i in eachindex(p)],
-    )
-end
-function get_basis_default(::DefaultManifold, p, N::RealNumbers)
-    return CachedBasis(
-        DefaultBasis(N),
-        [_euclidean_basis_vector(p, i) for i in eachindex(p)],
+for fname in [:get_basis_orthonormal, :get_basis_orthogonal, :get_basis_default]
+    BD = Dict(
+        :get_basis_orthonormal => :DefaultOrthonormalBasis,
+        :get_basis_orthogonal => :DefaultOrthogonalBasis,
+        :get_basis_default => :DefaultBasis,
     )
+    BT = BD[fname]
+    @eval function $fname(::DefaultManifold{ℝ}, p, N::RealNumbers)
+        return CachedBasis($BT(N), [_euclidean_basis_vector(p, i) for i in eachindex(p)])
+    end
+    @eval function $fname(::DefaultManifold{ℂ}, p, N::ComplexNumbers)
+        return CachedBasis(
+            $BT(N),
+            [_euclidean_basis_vector(p, i, real) for i in eachindex(p)],
+        )
+    end
 end
 function get_basis_diagonalizing(M::DefaultManifold, p, B::DiagonalizingOrthonormalBasis)
     vecs = get_vectors(M, p, get_basis(M, p, DefaultOrthonormalBasis()))
@@ -89,10 +88,10 @@ function get_basis_diagonalizing(M::DefaultManifold, p, B::DiagonalizingOrthonor
     return CachedBasis(B, DiagonalizingBasisData(B.frame_direction, eigenvalues, vecs))
 end
 
-# Complex manifold, real basis -> coefficients c are complesx -> reshape
+# Complex manifold, real basis -> coefficients c are complex -> reshape
 # Real manifold, real basis -> reshape
-function get_coordinates_orthonormal!(M::DefaultManifold, c, p, X, ::RealNumbers)
-    return copyto!(c, reshape(X, number_of_coordinates(M, ℝ)))
+function get_coordinates_orthonormal!(M::DefaultManifold, c, p, X, N::AbstractNumbers)
+    return copyto!(c, reshape(X, number_of_coordinates(M, N)))
 end
 function get_coordinates_diagonalizing!(
     M::DefaultManifold,
@@ -103,11 +102,11 @@ function get_coordinates_diagonalizing!(
 )
     return copyto!(c, reshape(X, number_of_coordinates(M, ℝ)))
 end
-function get_coordinates_orthonormal!(::DefaultManifold, c, p, X, ::ComplexNumbers)
+function get_coordinates_orthonormal!(::DefaultManifold{ℂ}, c, p, X, ::RealNumbers)
     m = length(X)
     return copyto!(c, [reshape(real(X), m); reshape(imag(X), m)])
 end
-function get_vector_orthonormal!(M::DefaultManifold, Y, p, c, ::RealNumbers)
+function get_vector_orthonormal!(M::DefaultManifold, Y, p, c, ::AbstractNumbers)
     return copyto!(Y, reshape(c, representation_size(M)))
 end
 function get_vector_diagonalizing!(
@@ -119,7 +118,7 @@ function get_vector_diagonalizing!(
 )
     return copyto!(Y, reshape(c, representation_size(M)))
 end
-function get_vector_orthonormal!(M::DefaultManifold{ℂ}, Y, p, c, ::ComplexNumbers)
+function get_vector_orthonormal!(M::DefaultManifold{ℂ}, Y, p, c, ::RealNumbers)
     n = div(length(c), 2)
     return copyto!(Y, reshape(c[1:n] + c[(n + 1):(2n)] * 1im, representation_size(M)))
 end
diff --git a/src/bases.jl b/src/bases.jl
index 53302a8b..aa37118e 100644
--- a/src/bases.jl
+++ b/src/bases.jl
@@ -48,7 +48,7 @@ Abstract type that represents a basis of vector space of type `VST` on a manifol
 a subset of it.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -63,7 +63,7 @@ An arbitrary basis of vector space of type `VST` on a manifold. This will usuall
 be the fastest basis available for a manifold.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -89,7 +89,7 @@ Abstract type that represents an orthonormal basis of vector space of type `VST`
 manifold or a subset of it.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -104,7 +104,7 @@ An arbitrary orthogonal basis of vector space of type `VST` on a manifold. This
 be the fastest orthogonal basis available for a manifold.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -137,7 +137,7 @@ Abstract type that represents an orthonormal basis of vector space of type `VST`
 manifold or a subset of it.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -153,7 +153,7 @@ An arbitrary orthonormal basis of vector space of type `VST` on a manifold. This
 be the fastest orthonormal basis available for a manifold.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # See also
 
@@ -184,7 +184,7 @@ of the ambient space projected onto the subspace representing the tangent space
 at a given point.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 Available methods:
   - `:gram_schmidt` uses a modified Gram-Schmidt orthonormalization.
@@ -205,6 +205,7 @@ end
 An orthonormal basis obtained from a basis.
 
 # Constructor
+
     GramSchmidtOrthonormalBasis(𝔽::AbstractNumbers = ℝ)
 """
 struct GramSchmidtOrthonormalBasis{𝔽} <: AbstractOrthonormalBasis{𝔽,TangentSpaceType} end
@@ -218,7 +219,7 @@ An orthonormal basis `Ξ` as a vector of tangent vectors (of length determined b
 tensor ``R(u,v)w`` and where the direction `frame_direction` ``v`` has curvature `0`.
 
 The type parameter `𝔽` denotes the [`AbstractNumbers`](@ref) that will be used
-for the vectors elements.
+as coefficients in linear combinations of the basis vectors.
 
 # Constructor
 
@@ -302,15 +303,15 @@ function allocate_result(
     f::typeof(get_coordinates),
     p,
     X,
-    basis::AbstractBasis,
-)
-    T = allocate_result_type(M, f, (p, X))
+    basis::AbstractBasis{𝔽},
+) where {𝔽}
+    T = coordinate_eltype(M, p, 𝔽)
     return allocate_coordinates(M, p, T, number_of_coordinates(M, basis))
 end
 
 @inline function allocate_result_type(
     M::AbstractManifold,
-    f::Union{typeof(get_coordinates),typeof(get_vector)},
+    f::typeof(get_vector),
     args::Tuple{Any,Vararg{Any}},
 )
     apf = allocation_promotion_function(M, f, args)
@@ -361,18 +362,17 @@ function combine_allocation_promotion_functions(::typeof(identity), ::typeof(com
 end
 
 """
-    coordinate_eltype(M::AbstractManifold{M𝔽}, p, 𝔽::AbstractNumbers) where {M𝔽}
+    coordinate_eltype(M::AbstractManifold, p, 𝔽::AbstractNumbers)
 
 Get the element type for 𝔽-field coordinates of the tangent space at a point `p` from
 manifold `M`. This default assumes that usually complex bases of complex manifolds have
 real coordinates but it can be overridden by a more specific method.
 """
-@inline function coordinate_eltype(::AbstractManifold{M𝔽}, p, 𝔽::AbstractNumbers) where {M𝔽}
-    if M𝔽 === 𝔽
-        return real(number_eltype(p))
-    else
-        return number_eltype(p)
-    end
+@inline function coordinate_eltype(::AbstractManifold, p, 𝔽::ComplexNumbers)
+    return complex(float(number_eltype(p)))
+end
+@inline function coordinate_eltype(::AbstractManifold, p, ::RealNumbers)
+    return real(float(number_eltype(p)))
 end
 
 @doc raw"""
@@ -406,15 +406,17 @@ function _dual_basis(
     return DefaultOrthonormalBasis{𝔽}(TangentSpaceType())
 end
 
-function _euclidean_basis_vector(p::StridedArray, i)
-    X = zero(p)
+# if `p` has complex eltype but you'd like to have real basis vectors,
+# you can pass `real` as a third argument to get that
+function _euclidean_basis_vector(p::StridedArray, i, eltype_transform = identity)
+    X = zeros(eltype_transform(eltype(p)), size(p)...)
     X[i] = 1
     return X
 end
-function _euclidean_basis_vector(p, i)
+function _euclidean_basis_vector(p, i, eltype_transform = identity)
     # when p is for example a SArray
-    X = similar(p)
-    copyto!(X, zero(p))
+    X = similar(p, eltype_transform(eltype(p)))
+    fill!(X, zero(eltype(X)))
     X[i] = 1
     return X
 end
@@ -559,8 +561,8 @@ function _get_coordinates(M::AbstractManifold, p, X, B::DefaultOrthonormalBasis)
     return get_coordinates_orthonormal(M, p, X, number_system(B))
 end
 function get_coordinates_orthonormal(M::AbstractManifold, p, X, N)
-    Y = allocate_result(M, get_coordinates, p, X, DefaultOrthonormalBasis(N))
-    return get_coordinates_orthonormal!(M, Y, p, X, N)
+    c = allocate_result(M, get_coordinates, p, X, DefaultOrthonormalBasis(N))
+    return get_coordinates_orthonormal!(M, c, p, X, N)
 end
 
 function _get_coordinates(M::AbstractManifold, p, X, B::DiagonalizingOrthonormalBasis)
@@ -572,8 +574,8 @@ function get_coordinates_diagonalizing(
     X,
     B::DiagonalizingOrthonormalBasis,
 )
-    Y = allocate_result(M, get_coordinates, p, X, B)
-    return get_coordinates_diagonalizing!(M, Y, p, X, B)
+    c = allocate_result(M, get_coordinates, p, X, B)
+    return get_coordinates_diagonalizing!(M, c, p, X, B)
 end
 
 function _get_coordinates(M::AbstractManifold, p, X, B::CachedBasis)
@@ -585,7 +587,7 @@ function get_coordinates_cached(
     p,
     X,
     B::CachedBasis,
-    ::RealNumbers,
+    ::ComplexNumbers,
 )
     return map(vb -> conj(inner(M, p, X, vb)), get_vectors(M, p, B))
 end
@@ -595,7 +597,7 @@ function get_coordinates_cached(
     p,
     X,
     C::CachedBasis,
-    ::𝔽,
+    ::RealNumbers,
 ) where {𝔽}
     return map(vb -> real(inner(M, p, X, vb)), get_vectors(M, p, C))
 end
@@ -644,7 +646,7 @@ function get_coordinates_cached!(
     p,
     X,
     B::CachedBasis,
-    ::RealNumbers,
+    ::ComplexNumbers,
 )
     map!(vb -> conj(inner(M, p, X, vb)), Y, get_vectors(M, p, B))
     return Y
@@ -656,7 +658,7 @@ function get_coordinates_cached!(
     p,
     X,
     C::CachedBasis,
-    ::𝔽,
+    ::RealNumbers,
 ) where {𝔽}
     map!(vb -> real(inner(M, p, X, vb)), Y, get_vectors(M, p, C))
     return Y
@@ -939,26 +941,22 @@ For array manifolds, this converts a vector representation of the tangent
 vector to an array representation. The [`vee`](@ref) map is the `hat` map's
 inverse.
 """
-@inline hat(M::AbstractManifold, p, X) =
-    get_vector(M, p, X, VeeOrthogonalBasis(number_system(M)))
-@inline hat!(M::AbstractManifold, Y, p, X) =
-    get_vector!(M, Y, p, X, VeeOrthogonalBasis(number_system(M)))
+@inline hat(M::AbstractManifold, p, X) = get_vector(M, p, X, VeeOrthogonalBasis(ℝ))
+@inline hat!(M::AbstractManifold, Y, p, X) = get_vector!(M, Y, p, X, VeeOrthogonalBasis(ℝ))
 
 """
-    number_of_coordinates(M::AbstractManifold{𝔽}, B::AbstractBasis)
-    number_of_coordinates(M::AbstractManifold{𝔽}, ::𝔾)
+    number_of_coordinates(M::AbstractManifold, B::AbstractBasis)
+    number_of_coordinates(M::AbstractManifold, ::𝔾)
 
 Compute the number of coordinates in basis of field type `𝔾` on a manifold `M`.
 This also corresponds to the number of vectors represented by `B`,
 or stored within `B` in case of a [`CachedBasis`](@ref).
 """
-function number_of_coordinates(M::AbstractManifold{𝔽}, ::AbstractBasis{𝔾}) where {𝔽,𝔾}
+function number_of_coordinates(M::AbstractManifold, ::AbstractBasis{𝔾}) where {𝔾}
     return number_of_coordinates(M, 𝔾)
 end
-function number_of_coordinates(M::AbstractManifold{𝔽}, f::𝔾) where {𝔽,𝔾}
-    # for odd manifolds this first case has to match.
-    (real_dimension(𝔽) == real_dimension(f)) && return manifold_dimension(M)
-    return div(manifold_dimension(M), real_dimension(𝔽)) * real_dimension(f)
+function number_of_coordinates(M::AbstractManifold, f::𝔾) where {𝔾}
+    return div(manifold_dimension(M), real_dimension(f))
 end
 
 """
@@ -1084,8 +1082,7 @@ For array manifolds, this converts an array representation of the tangent
 vector to a vector representation. The [`hat`](@ref) map is the `vee` map's
 inverse.
 """
-vee(M::AbstractManifold, p, X) =
-    get_coordinates(M, p, X, VeeOrthogonalBasis(number_system(M)))
+vee(M::AbstractManifold, p, X) = get_coordinates(M, p, X, VeeOrthogonalBasis(ℝ))
 function vee!(M::AbstractManifold, Y, p, X)
-    return get_coordinates!(M, Y, p, X, VeeOrthogonalBasis(number_system(M)))
+    return get_coordinates!(M, Y, p, X, VeeOrthogonalBasis(ℝ))
 end
diff --git a/test/ManifoldsBaseTestUtils.jl b/test/ManifoldsBaseTestUtils.jl
index fcdc9130..ce9fe2aa 100644
--- a/test/ManifoldsBaseTestUtils.jl
+++ b/test/ManifoldsBaseTestUtils.jl
@@ -306,7 +306,7 @@ function ManifoldsBase.exp!(
 end
 
 function ManifoldsBase.get_basis_orthonormal(
-    ::DefaultManifold,
+    ::DefaultManifold{ℝ},
     p::NonBroadcastBasisThing,
     𝔽::RealNumbers,
 )
@@ -319,7 +319,7 @@ function ManifoldsBase.get_basis_orthonormal(
     )
 end
 function ManifoldsBase.get_basis_orthogonal(
-    ::DefaultManifold,
+    ::DefaultManifold{ℝ},
     p::NonBroadcastBasisThing,
     𝔽::RealNumbers,
 )
@@ -332,7 +332,7 @@ function ManifoldsBase.get_basis_orthogonal(
     )
 end
 function ManifoldsBase.get_basis_default(
-    ::DefaultManifold,
+    ::DefaultManifold{ℝ},
     p::NonBroadcastBasisThing,
     N::ManifoldsBase.RealNumbers,
 )
diff --git a/test/allocation.jl b/test/allocation.jl
index 3838a20d..fdbfe172 100644
--- a/test/allocation.jl
+++ b/test/allocation.jl
@@ -95,4 +95,7 @@ ManifoldsBase.representation_size(::AllocManifold4) = (2, 3)
 
     @test default_type(M2) === Matrix{Float64}
     @test default_type(M2, TangentSpaceType()) === Matrix{Float64}
+
+    @test ManifoldsBase.coordinate_eltype(M, fill(true, 2, 3), ℝ) === Float64
+    @test ManifoldsBase.coordinate_eltype(AllocManifold4(), a4, ℍ) === QuaternionF64
 end
diff --git a/test/bases.jl b/test/bases.jl
index 49e2de2b..c40cf55e 100644
--- a/test/bases.jl
+++ b/test/bases.jl
@@ -219,14 +219,15 @@ using ManifoldsBaseTestUtils
         p = [1.0, 2.0im, 3.0]
         X = [1.2, 2.2im, 2.3im]
         b = [Matrix{Float64}(I, 3, 3)[:, i] for i in 1:3]
-        Bℝ = CachedBasis(DefaultOrthonormalBasis{ℝ}(), b)
+        bℂ = [b..., (b .* 1im)...]
+
+        Bℝ = CachedBasis(DefaultOrthonormalBasis{ℂ}(), b)
         aℝ = get_coordinates(M, p, X, Bℝ)
         Yℝ = get_vector(M, p, aℝ, Bℝ)
         @test Yℝ ≈ X
         @test ManifoldsBase.number_of_coordinates(M, Bℝ) == 3
 
-        bℂ = [b..., (b .* 1im)...]
-        Bℂ = CachedBasis(DefaultOrthonormalBasis{ℂ}(), bℂ)
+        Bℂ = CachedBasis(DefaultOrthonormalBasis{ℝ}(), bℂ)
         aℂ = get_coordinates(M, p, X, Bℂ)
         Yℂ = get_vector(M, p, aℂ, Bℂ)
         @test Yℂ ≈ X
@@ -380,23 +381,23 @@ using ManifoldsBaseTestUtils
         X = [2.0, 1.0im]
         Bc = DefaultOrthonormalBasis(ManifoldsBase.ℂ)
         CBc = get_basis(Mc, p, Bc)
-        @test CBc.data == [[1.0, 0.0], [0.0, 1.0], [1.0im, 0.0], [0.0, 1.0im]]
+        @test CBc.data == [[1.0, 0.0], [0.0, 1.0]]
         B = DefaultOrthonormalBasis(ManifoldsBase.ℝ)
         CB = get_basis(Mc, p, B)
-        @test CB.data == [[1.0, 0.0], [0.0, 1.0]]
-        @test get_coordinates(Mc, p, X, CBc) == [2.0, 0.0, 0.0, 1.0]
-        @test get_coordinates(Mc, p, X, CB) == [2.0, 1.0im]
+        @test CB.data == [[1.0, 0.0], [0.0, 1.0], [1.0im, 0.0], [0.0, 1.0im]]
+        @test get_coordinates(Mc, p, X, CB) == [2.0, 0.0, 0.0, 1.0]
+        @test get_coordinates(Mc, p, X, CBc) == [2.0, 1.0im]
         # ONB
         cc = zeros(4)
-        @test get_coordinates!(Mc, cc, p, X, Bc) == [2.0, 0.0, 0.0, 1.0]
+        @test get_coordinates!(Mc, cc, p, X, B) == [2.0, 0.0, 0.0, 1.0]
         @test cc == [2.0, 0.0, 0.0, 1.0]
         c = zeros(ComplexF64, 2)
-        @test get_coordinates!(Mc, c, p, X, B) == [2.0, 1.0im]
+        @test get_coordinates!(Mc, c, p, X, Bc) == [2.0, 1.0im]
         @test c == [2.0, 1.0im]
         # Cached
-        @test get_coordinates!(Mc, cc, p, X, CBc) == [2.0, 0.0, 0.0, 1.0]
+        @test get_coordinates!(Mc, cc, p, X, CB) == [2.0, 0.0, 0.0, 1.0]
         @test cc == [2.0, 0.0, 0.0, 1.0]
-        @test get_coordinates!(Mc, c, p, X, CB) == [2.0, 1.0im]
+        @test get_coordinates!(Mc, c, p, X, CBc) == [2.0, 1.0im]
         @test c == [2.0, 1.0im]
 
     end
diff --git a/test/default_manifold.jl b/test/default_manifold.jl
index ef1c0c86..44bc97d7 100644
--- a/test/default_manifold.jl
+++ b/test/default_manifold.jl
@@ -636,11 +636,19 @@ using ManifoldsBaseTestUtils
         MC = ManifoldsBase.DefaultManifold(3; field = ManifoldsBase.ℂ)
         p = [1.0im, 2.0im, -1.0im]
         CB = get_basis(MC, p, DefaultOrthonormalBasis(ManifoldsBase.ℂ))
-        @test CB.data isa Vector{Vector{ComplexF64}}
-        @test ManifoldsBase.coordinate_eltype(MC, p, ManifoldsBase.ℂ) === Float64
-        @test ManifoldsBase.coordinate_eltype(MC, p, ManifoldsBase.ℝ) === ComplexF64
+        @test CB.data == [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]
+        @test CB.data isa Vector{Vector{Float64}}
+        @test ManifoldsBase.coordinate_eltype(MC, p, ManifoldsBase.ℂ) === ComplexF64
+        @test ManifoldsBase.coordinate_eltype(MC, p, ManifoldsBase.ℝ) === Float64
         CBR = get_basis(MC, p, DefaultOrthonormalBasis())
-        @test CBR.data == [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0], [0.0, 0.0, 1.0]]
+        @test CBR.data == [
+            [1.0 + 0.0im, 0.0 + 0.0im, 0.0 + 0.0im],
+            [0.0 + 0.0im, 1.0 + 0.0im, 0.0 + 0.0im],
+            [0.0 + 0.0im, 0.0 + 0.0im, 1.0 + 0.0im],
+            [0.0 + 1.0im, 0.0 + 0.0im, 0.0 + 0.0im],
+            [0.0 + 0.0im, 0.0 + 1.0im, 0.0 + 0.0im],
+            [0.0 + 0.0im, 0.0 + 0.0im, 0.0 + 1.0im],
+        ]
     end
 
     @testset "Show methods" begin
@@ -671,7 +679,7 @@ using ManifoldsBaseTestUtils
     end
 
     @testset "scalars" begin
-        M = DefaultManifold()
+        M = ManifoldsBase.DefaultManifold()
         p = 1.0
         X = 2.0
         @test copy(M, p) === p