What’s the big deal? Well for me the fascination was that something so complex as the Mandebrot set could be represented by such a simple-seeming equation: z = z2 + c. I think the beautiful imagery that arises from such a simple equation explains its enduring appeal. It is one of the things that really inspired me to get into more math as a kid (such inspiration was of course much evaporated by the time I got to college).
A very interesting modern variation is called the “Buddhabrot”:
The Buddhabrot is featured on complexification.net, a fascinating “gallery of computation” focusing on very cool mathematical eye candy, including some algorithms that are featured in the wonderful xscreensaver program. Xscreensaver is the evil eye candy screensaver that runs on my Linux machines – the program is a fantastic gallery of fun algorithmic displays put together by Jamie Zawinski, one of the original Netscape luminaries (he helped write v1.0 of the Netscape browser for Unix – nowadays he seems to have given up on software altogether by running a nightclub in San Francisco).