> The C that shows up in quantum mechanics is likely an example of this--it's a case of physics having a a circular symmetry embedded in it (the phase of the wave functions) and everyone getting attached to their favorite way of writing it
No, it really is C, not R^2. Consider product spaces, for example. C^2 ⊗ C^2 is C^4 = R^8, but R^4 ⊗ R^4 is R^16 - twice as large. So you get a ton of extra degrees of freedom with no physical meaning. You can quotient them out identifying physically equivalent states - but this is just the ordinary construction of the complex numbers as R^2/(x^2 + 1).
> but rather, physics uses C because C models the algebra of the thing physics is describing.
That's what C is: R^2, with extra algebraic structure.
Yes I know and agree with that. But still I think physics can be described with either. There will, I expect be a physical meaning to that quotient. Maybe the larger space without the quotient is also physically meaningful too.
No, it really is C, not R^2. Consider product spaces, for example. C^2 ⊗ C^2 is C^4 = R^8, but R^4 ⊗ R^4 is R^16 - twice as large. So you get a ton of extra degrees of freedom with no physical meaning. You can quotient them out identifying physically equivalent states - but this is just the ordinary construction of the complex numbers as R^2/(x^2 + 1).
> but rather, physics uses C because C models the algebra of the thing physics is describing.
That's what C is: R^2, with extra algebraic structure.