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Fractal plotting for the collatz fractal

I have the following equation

f[z_] := 1/4 (2 + 7 z - (2 + 5 z) Cos[Pi*z])

I want to map this on the imaginary plane such that if it converges under iterations of the function then it is plotted. I would like to then color this in a Mandelbrot type way. How would i go about plotting this?

The goal is to explore the fractal mentioned here

This is the type of image i'm tring to produce with the ability to zoom in to particular points.

Goal