Ultra Fractal.

When you are ready for more power, Ultra Fractal is an awesome, professional package that supports animation, arbitrarily deep fractals, extremely high resolution, multiple layers, transparency, 3D and so many other neat things that there just is no way I can tell you about all of them right here. I will how ever just point out a few basic but necessary things that I think are neat about ultra fractal.

1) You are able to create fractals at resolutions much larger than your desktop.

work in resolutions larger than your desktop

2) There is a neat little switch mode that allows you to preview the corresponding Julia set in the sidebar

switch

3) You can link opacity and color within your color palettes on each layer.

Opacity

4) there is a formula editor, allowing you to adjust and play around with existing formulas.

formula-editor

There is a database of fractals you can download for free, giving you thousands of fractals to explore.
All in all I found ultra fractal very straight forward and easy to use.
It has excellent tutorials for beginners and many of the functions and options that are available to you as an artist or a fractal explorer are almost overwhelming in choice – allowing you to create a project that is as simple or as complicated as you want.
Ultra Fractal.