Some thoughts on the form Zn=(Z(n-1))^2+C:

^2 can be replaced with ^P, where P is any number greater than 1. Whole numbers produce sets with P-1 buds, whereas other numbers produce extremely interesting chaotic sets (P=1.5 is my personal favorite). P can also be complex, which produces even cooler fractals; these don't even have the characteristic set centered around the origin!

Multiple dimensional fractals can be made by having any combination of Zr, Zi, Cr, Ci, Pr, and Pi be dependent on any number of axis (four is the most human minds can usually handle ;). I have not yet tested quaternions or octonions.

If any point Z0 is found to be not contained within the set, then any of the Zn's encountered during iteration are also not contained. The opposite is theoretically true, but only if you iterate infinitely.