One of my favorite toys when it comes to computing is an absolutely gorgeous application called ChaoScope, a tool for visualising and creating high quality images of these incredible systems.
Here is a small rendering I created of the Julia set in quaternion space:

It's possible to spend hours at a time getting lost in the world that this application lets you into. Download it and play for a while and, if you need your programs to have a "purpose" then, I can promise you that you'll never be at a loss for a desktop background again.