These fractals are based on hypercomplex numbers, which like quaternions are a four dimensional generalization of complex numbers. It is not possible to fully generalize the complex numbers to four dimensions without sacrificing some of the algebraic properties shared by real and complex numbers. This law holds most of the time but not all the time for hypercomplex numbers. However hypercomplex numbers have a wonderful property for fractal purposes.
Every function defined for complex numbers has a simple generalization to hypercomplex numbers.
You can see 3D versions of these fractals using fractal type Julibrot. See also Quaternion and Quaternion and Hypercomplex Algebra Back to The Fractint Home Page.