There are several GeoGebra applets which demonstrate the Mandelbrot set, that is, the subset of the complex plane of numbers z such that  is bounded. One of my favorites is Daniel Piercy's work "Painting the Mandelbrot Set". The original one is a little bit slow in a browser, so I minimized and modified it to work faster.

Here the magenta points are elements of the Mandelbrot set, black points are outside, blue points are at the boundary. Another approach is the following one which was copied from Mike Borcherds's collection:

Here the black background points are elements of the set, and yellow-green points are outside, other colors mean the boundary. The draggable red point shows the orbit of z in the formula above. For the best experience perhaps it's even better to check out the XaoS application which very efficiently computes and zooms in the Mandelbrot set. Its zooming engine has also been implemented in pure JavaScript by J. B. Langston.