Yet another way to visually understand complex functions is to think them as 2D plane to 2D plane transformations. You draw a picture, grid or curves into complex plane, then run it trough complex function you are interested in and see how it looks like.
Here is w = e^z:
https://i.imgur.com/pAALOh2.png
Just look at the above picture and every detail until it starts to makes sense.