I was never satified with the visualisations of functions with complex variable and complex value.To show 4 dimensions on a flat screen is not easy, but nobody tried it hard enough.
One approach is to show the absolut value as height and the argument as color,
another approach was to show both dimension of the value in a color coding, what is not bad, but to recognize the shape from a color gradient is really difficult.

In my approach we can show 3 dimensions without problem, and the 4-th one i also code as color.
Still one dimension color coded is not easy to understand, so at least we can choose which one showed as height, and which as color. If our brain can bring both pictures together (still difficult), a deeper understanding of the complex function become possible.

For example if you check the two viewes of sine, you will recognize the close relation between trigonometric and hyperbolic functions visually, what you may know already from their formulas.

A detailed viewer for an arbitrary complex function you can find under www.complex.jampas.info.
Here you can see some demos recorded from an older version...