When dealing with Mach-Zender interferometers the professor usually lets $\alpha$ & $\beta$ denote the probability amplitude that a particular photon isn't reflected by the beam splitter and the probability amplitude that a particular photon is reflected by the beam splitter respectively.
What exactly is probability amplitude? Supposedly the actual probability that a photon is not reflected by the beam splitter is $|\alpha |^2$, similarly with $\beta$, so that $|\alpha |^2 + |\beta|^2 = 1$.
Why are $\alpha$ and $\beta$ complex numbers? The professor explains it by pointing out to the fact that the wave function is complex in nature, but that just begs the question: Why is the wave function complex in nature?
To be clear, I understand why it is complex mathematically, I'm rather asking for the physical phenomena that requires us to work with complex numbers instead of real ones.