Efficient complex differentiation by structured 2x2 matrices

For efficient complex differentiation, we need to express the following structured matrices:

holomorphic:

[a -b]
[b  a]

anti-holomorphic:

[a  b]
[b -a]

C->R:

[a b]
[0 0]

R->C:

[0 0]
[a b]

general:

[a c]
[b d]

Wirtinger derivative archives this by doing a basis transformation from x, y to z, z̄. However, that would introduce more FLOPs, since you need to transform them to x, y when multiplying a complex number. I.e. (x = z+z̄) and (y = i(z - z̄))

IMO, structured matrices are far more transparent than Wirtinger derivative, and they don't require a change of basis before multiplying with a complex number.

To implement structured matrices, we could do

struct Holomorphic{T,S}
    a::T
    b::S
end

Base.:*(::Holomorphic, ::Complex) = ...

Start by reviewing the proposed Holomorphic type and the Base.:* dispatch with Complex values, then compare the listed holomorphic, anti-holomorphic, C->R, R->C, and general matrix forms. The issue does not name files or tests; done requires an agreed structured-matrix design and coverage for the listed forms without the proposed basis transformation.