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I was fiddling with this complex function visualizer and accidentally found this function which looks a lot like the blackhole visualizations that I see on the net: $$ f(z)=(z\bar{z}-1)^z $$ and I'm curious if there is a real connection with physics or is it just pure coincidence?

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This is not connected to black holes at all. Complex numbers are 2D objects to you need to be careful when looking at those visualizations as they are trying to represent a mapping from 2D onto 2D, which by definition requires a 4D space.

So you are seeing some form of incomplete picture of the mapping and the page does not explain what it is showing. Very commonly these images show a different color for e.g. the number of times the expression can be iterated on a particular value of $z$ before the argument of $z$ exceeds some threshold (typically $1$). This is how, e.g. Mandelbrot maps are made. This is one form of domain coloring.

The function itself $f(z)$ has no "singularity-like" features and seems to be a well behaved and continuous function from what I can quickly tell. If you're unused to manipulating complex numbers it's a good exercise to use the polar form of $z$ and De Moivre's formula to work out what $f(z)$ is in polar form.

No black holes here. :-)

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