These fractals are based on quaternions. Quaternions are an extension of complex numbers, with 4 parts instead of 2.
Quaternions have rules for addition and multiplication.
The normal Mandelbrot and Julia formulas can be generalized to use quaternions instead of complex numbers. There is one complication. Complex numbers have 2 parts, so they can be displayed on a plane. Quaternions have 4 parts, so they require 4 dimensions to view. That is, the quaternion Mandelbrot set is actually a 4-dimensional object.