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The ontological question of what the wave function means is independent, it seems to me, of whether or not it uses complex numbers. If you had a physical interpretation that said something like "the wave function is a length of some object" or "the wave function is a probability", then certainly complex values are unphysical. But it seems to me that worrying about whether it's complex prior to coming up with a physical meaning for it is putting the cart before the horse.The question is a philosophical one: why did Nature "choose" complex amplitudes over real ones?

Certainly, you can always eliminate complex numbers by using matrices instead of numbers, or by using quaternions, or whatever. But I don't see that such a change is anything more than aesthetic.

On the other hand, Hestenes had a program, spacetime algebra, for replacing all occurrences of complex numbers in physics by geometric objects such as elements of a Clifford algebra. To a large extent, it can be done. I'm not sure, though, that it actually has helped in figuring out the ontology of the wave function. Hestene's interpretation of the Schrodinger equation, for example, interprets the ##i## in ##H |\psi\rangle = i \frac{d}{dt} |\psi\rangle## not as an imaginary number, but as a bivector representing the spin of the particle. That makes the Schrodinger equation into an approximation to the Pauli equation. So it's sort of interesting.