[diedyesterday]

I don't think complex numbers are entirely without "fundamental physical basis"; Distinct antiparticles (antimatter) could be said to be the manifestation of complex numbers in the physical world. The important thing to keep in mind about complex numbers is that they have 2 degrees-of-freedom (DoFs); Analogously if a quantum field has two DoFs (that couple in a certain way) it is said to be a complex field and it will have two kinds of distinct particles.

Real-valued (operator) fields (like the photon field) do not have distinct particles (a photon is its own antiparticle)

I can elaborate more if needed.