What is probability amplitude and why is it complex? 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.


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*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.
 A: You've answered (1) correctly. A probability amplitude is that thing that you take the square of (well, absolute value squared) to get probability.
That leaves (2), which is essentially the question "why does QM use complex numbers?". 
I'm not sure. It's certainly the case that this is just a mathematical choice and convention - you can write QM with only Real numbers. The theory will just be uglier.
So in a way, complex numbers are only needed to keep the theory neat. Linear. A linear algebra. QM is phrased in a simple and clear manner when using the language of complex numbers, so we phrase it like that. 
Note that the complex phase (angle of the probability amplitude as a complex number) of the state has no importance in of itself. It's only relative phases, between states, that matter. So the fact that we're using complex numbers is connected to how states can be related to each other. 
For example, we find experimentally that in nature we can distinguish between states being in a state like $|+\rangle=1/\sqrt{2}(|0\rangle + |1\rangle)$ as opposed to $|i\rangle=1/\sqrt{2}(|0\rangle+i|1\rangle)$. E.g. measurement in the $(+,-)$ basis will return + 100% of the time if the system is in state $|+\rangle$ , but we'll get + only 50% of the time if that system is in state $|i\rangle$.
So we know there is a structure to how states can be related, which corresponds to the structure of complex numbers. 
A: 
  
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*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 $|\alpha |^2 + |\beta|^2 = 1$.
  

As you said, the probability amplitude is a complex number whose squared magnitude is the probability that the system will be in that state. The reason we care about this amplitude is because it's the coefficient on the state in the wavefunction, so we can read it off immediately from the wavefunction itself.


  
*Why are $\alpha$ & $\beta$ complex numbers? The professor explains it by pointing out to the fact that the Wavefunction is complex in nature, but that just begs the question: Why is the Wavefunction complex in nature?
  

You can represent the wavefunction in other ways that don't involve complex numbers at all. The simplest example comes from the fact that you can represent every complex number as a matrix of real numbers. In particular, the number $a+bi$ is equivalent to the matrix $\begin{pmatrix}a&-b\\b&a\end{pmatrix}$. Using this, every complex wavefunction can be expressed as a matrix of real wavefunctions.
The larger point of this example is that the wavefunction requires a more complicated mathematical structure than is possible with a single real function, and complex numbers have been conventionally chosen as the easiest way to represent such a structure. You can certainly get rid of the complex numbers if you don't like them, but you can't get rid of the more complicated algebraic structure that prompts their use.
