I am yet to be convinced there is sufficient need to expose complex numbers. Maybe some exposed functions can return a tuple with real and imaginary parts (perhaps not in the transformed parameters or model blocks), but I think most useful things can be implemented internally in Stan Math.
I am in agreement - unless someone implements a full complex algebra everything will be piecemeal implementations that are more appropriately wrapped in the function for which the partial complex algebra implementation is needed. In that case we should be able to get away with functions that return real outputs (i.e. power and phase functions from an FFT, magnitude and phase functions for input complex numbers, etc) instead of a partial complex algebra implementation that is going to be extraordinarily confusing to users.
On the original question, when I asked for this, I just wanted the largest eigenvalue for certain nonsymmetric matrices. This was actually for cases where you can guarantee they are real, but it seems you cannot compute them without complex numbers.