It is difficult to separate the mathematics from the art, but what makes fractals appealing to me is the shapes and forms within them, some beautifully complex and infinitely detailed, others much more simple or fun, or curiously mimicking things found in everyday life. That these images were created with a mathematical formula does not detract from their beauty or invalidate it. The argument of whether fractals are art is an old one. Many people argue that anything created by a computer cannot be considered art, and that all that is required to create fractal images is the tapping in of a few numbers. In my view that is akin to saying that photographic art is achieved simply by the pressing of a button.
Fractal artists are explorers of strange worlds.
The computer and software they use are simply their tools of trade, that enable them to record what they create and find within those worlds. In much the same way that a photographer may come across an image or scene he likes, or have something in his mind's eye that he wants to recreate on paper, a fractal artist does the same but works with complex and imaginary numbers, experimenting with formulas and variables to achieve different effects, zooming and manipulating, and applying different colour schemes to achieve the same end. The fractal formulas are readily available or new ones can be written, but good fractal images don't make themselves any more than good photos do. It takes some creativity and intuition, and a lot of time and patience to create fractal images.