Before we go on, you need to understand the idea of a prisoner set. Basically, these sets have a function that if a number gets trapped in a pattern of numbers and is unable to escape to infinity, it becomes a prisoner.
The most notable person in the area of prisoner sets was a man by the name of Gaston Julia. His collection of Julia sets (appropriately named) were based upon the function f(z) = z^2 + c, in which changing the constant c produced a different Julia set. Now, some Julia sets were connected and others were not. Julia, however, had no way of determining whether a set was connected or not without iterating a series of points through that particular function.
Did I forget to mention that we are in the complex number system? Basically, you can look at a complex number a+bi as coordinates (a,b) if it will help you understand this any better.
Well, there are indications of self-similarity if you go searching for it, but this is only near self-similarity. Moreover, the Mandelbrot set does not the requirement of non-integral complexity, so it is not a fractal. The complexity of the set is 2 dimensions.