Concept for Mandelbrot set render in 3D:
Interior disc-like hyperbolic components are made 3D sphere-like balls, cardioid-like also made 3D in a similar way. Use a transparent material like glass.
Atom domain boundaries outside the Mandelbrot set (mostly circle-like) are made into 3D sphere-like balls using a thin-film soap bubble material.
The opaque ground plane intersects the structure, and is coloured in a monochrome fractal checkerboard pattern using bitwise XOR of X and Y coordinates.