Complex variable derivatives – Mathematics Stack Exchange

I’ve been trying to understand the complex function derivatives, but there are some issues that I’m encountering. I’m following the Ahlfors Complex Analysis. When he is demonstrating the derivative via the limit, he does it both with a pure infinitesimal Real and a pure Imaginary value, but I don’t get why should it be concluded that they must have the same value since you are going in different directions of the complex plane (with the real value you are going in the real axis and with the imaginary value you are going with the imaginary axis).

Also, I see some similarities with the multivariate calculus, without being the same, and that might be something that is causing me to some misconception.