Visualizations of functions in the split-complex/hyperbolic plane:
Mandelbrot (2nd power)
Mandelbrot (4th power & magnified)
Mandelbrot (5th power)
Visualizations of functions in the complex hexapolar plane:
These GIFs depict orthant-grouping logic conditionally applied to subsets of orthants for symmetry preservation/presentation:
Hexapolar Mandelbrot variations visualized with exponents 2 through 11: click to enlarge
Hexapolar Tricorn/Mandelbar variations visualized with exponents 2 through 11: click to enlarge
These GIFs depict orthants grouped by product with integer exponent of i without exception for ease of generalizing from the case of the complex plane:
Hexapolar Mandelbrot variations visualized with exponents 2 through 11: click to enlarge
Hexapolar Tricorn/Mandelbar variations visualized with exponents 2 through 11: click to enlarge
Hyperland Open Science Framework 2013 (I have been credited with inventing the 4D visualization mode demonstrated in the video below.)
I have enjoyed computational exploration of higher-dimensional and multipolar spaces. (Here is a fun simulator of central force potentials by Wolfgang Christian.) Here is a draft overview of a number system that is the minimum nontrivial multipolar embedding of the complex plane and the real line it contains; here is a more in-depth draft regarding multipolars, generalized inner product operations, and algebraic structures. (Please let me know if your are aware of relevant precedent ancedotally or in the literature: ben at ben blohowiak dot com.)