The issue is not the "imaginary numbers not real etc!!" fluff. Maths is a tool to model reality and perhaps imaginary/real is a bad name. The interesting questi

The issue is not the "imaginary numbers not real etc!!" fluff. Maths is a tool to model reality and perhaps imaginary/real is a bad name. The interesting question why is C the correct choice of mathematical field to do "quantum mechanics" over -- what are the physical reasons this is needed. You do wonder what exactly is meant by "doing quantum mechanics". I assume the start of the story is going from functions over C, quantising and getting some Hilbert space of operators/ getting wavefunctions. Naively I would think that "quantum mechanics over the reals" might be less interesting somehow. Consider the difference between doing complex analysis and real analysis. C loses total ordering as a field, but then gains additional nice properties over R. Quaternions however are non commutative which is a bit strange. Complex differentiable functions are more restrictive than functions on RxR (need to satisfy a stronger property). At some level you would think as models of reality you might need one structure over the other.