Undiscovered fractal sets?

Question

I'm interested in the topic of fractals, such as those created by the borders of the Mandelbrot and Julia sets.

My question is if there are other, not yet discovered fractal sets, which one could find by experimenting with the generation formulas (using a computer). I mean of course fractals which are substantially different in their appearance and "behaviour" than the ones which are already known.

Answer 1

The following links give some general information about several types of fractals:

http://www.fractalsciencekit.com/types/types.htm

Here are examples of many different types of fractals:

http://www.fractalsciencekit.com/gallery/gallery.htm

Answer 2

There exits other type of fractals which are generate using no only one function like in the clasic sense, we can generate fractal for a semigroup of rational functions, this was my undergraduate thesis, here some images.