My professor argues that one of the fundamentally unique properties of Quantum Mechanics is that the imaginary unit i is not removable (you can't avoid using it, unlike in other areas of physics like AC circuits where it is merely a convenience).
However, I'm not convinced. Isn't the reason i appears in the schrodinger equation that the equation needs to relate the wavefunction to its derivative? For convenience, we represent the wavefunction as a (possibly infinite) linear combination of complex exponentials, and this makes taking the derivative extremely easy which is where the factor of i comes from. However, can't complex exponentials always be replaced by sines and cosines? Wouldn't there be some way to state schrodinger's equation without using i as a result? The i merely represents the phase shift between the wavefunction and its derivative, so it makes the schrodinger equation much easier to write down (it would be more difficult to write a differential equation with an explicit phase shift). However, I don't see why a version of the schrodinger equation communicating the phase shift directly rather than by using i would be any less correct. Can you provide an argument for why the use of complex numbers is essential in quantum mechanics?