Another fun fractal that I like is the Julia Set. What I've recently discovered thanks to someone I was talking to (sorry, can't remember his name) is that there is a section of the Mandelbrot zoom that looks eerily similar to the Julia Set. Look at the Comparison below.
Here's a Julia Set that I generated using GIMP2.
And here's a Mandelbrot zoom generated (by the above mentioned anonymous person) using Fractal eXtreme. (Much nicer quality than my GIMP rendering)
Pretty neat! No?
If you look closely, you'll see that they are different. If you can't see it, look at the center. Here's the Wikipedia explanation.
At first sight, these islands seem to consist of infinitely many parts like Cantor sets, as is actually the case for the corresponding Julia set Jc. Here they are connected by tiny structures so that the whole represents a simply connected set. These tiny structures meet each other at a satellite in the center that is too small to be recognized at this magnification. The value of c for the corresponding Jc is not that of the image center but, relative to the main body of the Mandelbrot set, has the same position as the center of this image relative to the satellite shown in zoom step 7.
While we're on the subject of Mandelbrot vs. Julia, I'll leave you with a Jonathan Coulton fan video of my favorite JoCo song--pay extra special attention to the sound clip at the very end ;-)