I have seen the z transform here and there, but to be honest it confuses me the second I stop using it and start trying to understand it. It is sometimes regarded as the time delay operator for discrete signals
$$x[n-1] = z^{-1}x[n]$$
and sometimes as a complex value
$$X(z) = \sum_{n=-\infty}^{+\infty}x(n)z^{-n}$$
for which we analyze a region of convergence and such.
Now, I think I understand the first one decently in the context of LTI systems. The second I interpret as a generalization of the discrete Laplace transform. But I have a really hard time connecting the two! Why do we call a time delay operator a complex value?
Thanks in advance!
The first equation you write makes no sense at all, as you are mixing the time-domain and $\mathcal{Z}$-domain representation of a signal.
You probably got the idea that equation "holds" by looking at block diagrams where a signal $x[n]$ enters a "box" labeled $z^{-1}$ and the output is $x[n-1]$. Clearly, what the "box" does is to introduce a delay of single unit to its input. Notationally, it would be more appropriate to label this box as $\delta[n-1]$, suggesting that the box represents an LTI system with impulse response $h[n]=\delta[n-1]$ and, therefore, with an input $x[n]$, the output is $x[n]*h[n]=x[n]*\delta[n-1]=x[n-1]$. However, it is more convenient and has become the norm to write instead $z^{-1}$, which is the $\mathcal{Z}$-transform of $h[n]=\delta[n-1]$, i.e., $H(z)=z^{-1}$.