Where does the information in a fractal come from? When I view a fractal such as the Mandelbrot, my first thought is, where did this interesting picture come from. For a picture of this complexity, the information that generated this picture must be stored somewhere. Then I found that the Mandelbrot is generated by computational processes based upon mathematics in the area of complex numbers.

However, my question still holds, where did the information come from? The nice picture wasn't input when the program was written, where did the information for this picture come from?
To a large extent, the answer will depend on exactly what you mean by "information". The fractal has very low Kolmogorov complexity, as there is a very simple rule describing how you generate it. So in one sense, it contains very little information. On the other hand, a picture of such a fractal appears very complicated, even though there is a simple rule underlying it. This is not at all uncommon. Many times very simple rules give rise to very complex behavior. Examples of this range from cellular automata and Conway's game of life to things like the apparent complexity in the digits of $\pi$. The fact is that this new representation doesn't contain anything fundamentally new, but is rather a less efficient expression of the same object.