About the Collatz Conjecture, Complex Maps and Holomorphic Functions

First of all, I wasn't quite sure of where to post this thread, if it is wrongly located please forgive me.

Just out of curiosity I've been pondering over some holomorphic functions, for generating fractals, associated with the Collatz Conjecture. Now I've come up with these complex maps which I don't really understand that well. I got something out of skimming through Wikpedia articles about Holomorphic function Conformal map, but I'm hoping that someone would have more intuitive way of explaining how to read these diagrams, or maybe someone could point me towards a good website.

I would highly appreciate if someone could explain this concept with reference to the function

$f(z)=(1+(-1)^z)(z/4)+(1-(-1)^z)((3z+1)/2)$

The complex map generated by this function:

I go to bed now and check if I've got any replies tomorrow morning (that is four to five hours from now).

Thank you.