dot(::Adjoint, ::Adjoint) incorrect for numbers that are not multiplicatively commutative

[sethaxen]

The overload dot(x::Adjoint, y::Adjoint) assumes that the number eltypes of the inputs commute under multiplication, so that dot(x', y') == sum(x .* y') == conj(sum(x' .* y)) == conj(dot(x, y))
https://github.com/JuliaLang/julia/blob/1aac07efe44c229abf41b9c50031783f46bed4bf/stdlib/LinearAlgebra/src/generic.jl#L889

For numbers that don't commute under multiplication (like Quaternions), this definition is incorrect. A short demo:

julia> using Quaternions, LinearAlgebra

julia> mydot(x, y) = mapreduce(mydot, +, x, y);  # define our own recursive dot function

julia> mydot(x::Number, y::Number) = dot(x, y)

julia> x = [quatrand() for _ in 1:2, _ in 1:2];

julia> y = [quatrand() for _ in 1:2, _ in 1:2];

julia> dot(x, y) ≈ mydot(x, y)  # fine
true

julia> dot(transpose(x), transpose(y)) ≈ mydot(transpose(x), transpose(y))   # fine
true

julia> dot(x', y') ≈ mydot(x', y')   # uh-oh
false

julia> method = @which dot(x', y')
dot(x::Adjoint, y::Adjoint) in LinearAlgebra at /Applications/Julia-1.7.app/Contents/Resources/julia/share/julia/stdlib/v1.7/LinearAlgebra/src/generic.jl:922

julia> Base.delete_method(method)

julia> dot(x', y') ≈ mydot(x', y')   # all good
true

This could be handled by restricting the types to Union{Real,Complex}, e.g.

 dot(x::Adjoint{Union{Real,Complex}}, y::Adjoint{Union{Real,Complex}}) = conj(dot(parent(x), parent(y)))