Complex numbers have a tendency to spring up in more advanced physics concepts such as electrical engineering, more sophisticated versions of refraction and of course in the waveform of particles in quantum mechanics. It is simply very useful to have a whole mathematical dimension off the real number line to work with when dealing with things of varying phase relation. The mathematical explanation of why we can’t just use a basic sinusoidal function for an electron’s wave is because such a result would give absolute solutions when we know that there has to be uncertainty in the answer to the wave equation.
But of course this is using experimental observations to validate our mathematical assumption. There is no inherent physical reason why complex numbers should be selected over real numbers or the even more confusing hypercomplex numbers. Real numbers are shown to be insufficient using the logic above but whether hypercomplex numbers are applicable or, more importantly, required, has still yet to be answered. An important part of hypercomplex numbers is their lack of commutativity (the order of multiplication matters: a × b ≠ b × a) and this would carry over into hypercomplex quantum theory. Of course the basic way to test such a theory is to use the mathematics to make a unique prediction and then if that prediction is shown experimentally to be true, that is pretty good evidence for the theory. The prediction here is that the noncommutativity should result in the superposition of the phases of the particle waves not commuting either.
There have been attempts to observe such an effect in the past but the particles used were massive and slow moving meaning the quantum effects were heavily dampened. Now with modern engineering of metamaterials a much more sophisticated version of the experiment has been performed by passing a single photon through a negative refractive index material while completely contained in a path interferometer. It was found, to a high degree of precision, that the phases did indeed commute when the split was reunited. This doesn’t necessarily mean that hypercomplex numbers are an invalid way of looking at quantum systems but it does mean that the limits of where such a theory could work have been reduced and it may be that further experiments only reduce this range even further.