The Mandelbrot set is already 4-dimensional, sort of. It's called a Julibrot or Juliabrot fractal. In the definition of Mandelbrot and Julia sets (which use the

The Mandelbrot set is already 4-dimensional, sort of. It's called a Julibrot or Juliabrot fractal. In the definition of Mandelbrot and Julia sets (which use the same formula), Mandelbrot maps C to the output axes and Z starts at zero. For Julia, it's the other way around, but C is some constant rather than always zero, so there are lots of julia sets (although not all are interesting). If you combine these and make both Z and C variables, you go from 2 dimensions to 4. If you count the escape iterations, that gives you 5. This is basically the set of all Julia fractals, which also happens to contain the original Mandelbrot set across the hyperplane where Z=0.