Since I am currently studying for a Analysis exam and have always been fascinated by fractals, I wrote a small Mandelbrot set visualization in Python.
The core formula is the series of z = z^2 + c.
It was implemented with numpy complex and the pillow image library.
To increase performance this could be implemented in GLSL, since the operation is easily separable to multiple threads. For example one thread per pixel, as available in the fragment shader.