All of the animations produced so far have been of cubic quaternion julia sets
.
This is but a small set of the potential maps.
One possibility is to go to a cubic map
or higher.
I have written an iteration library which will iterate many functions of this form and others, not being limited to quaternions. An interesting system to explore is the complex Mandelbrot/julia system, where for each point on the complex plane we have another plane which is the julia set parameterized by the c value at that point. This is a four dimensional system which can be visualized in the same manner as quaternion fractals. One can, for instance, visualize all of the julia sets taken from some arbitrary line in the Mandelbrot set.