The Mandelbrot Set is built around a simple iterative equation.
z(1) = c z(n+1) = z(n)^2 + c
For any complex c, we can continue this iteration until either abs(z(n+1)) > 2 or n == lim, then return the iteration count n.
For a matrix of complex numbers C, return a corresponding matrix N such that each element of N is the iteration count n for each complex number c in the matrix C, subject to the iteration count limit of lim.
If C = [0 0.5; 1 4] and lim = 5, then N = [5 4; 2 1]
Cleve Moler has a whole chapter on the Mandelbrot set in his book Experiments with MATLAB: Chapter 10, Mandelbrot Set (PDF)