I know very little about the Mandelbrot set or complex dynamics, but there are several features which strongly suggest that its boundary is not the attractor of an iterated function system, at least a reasonable one.
For example.
- Around Misiurewicz points (which are only countable but dense), the Mandelbrot set looks locally like the corresponding Julia set, in particular it looks locally very different for each Misiurewicz point. Under some natural assumptions, sets invariant under IFS's look locally like the set itself.
- If an attractor of an IFS is connected, it is locally connected. The boundary of the Mandelbrot set is connected but it is a famous open problem whether it is locally connected.
- Under some weak contractivity assumptions on the generating maps, attractors of IFS's cannot have full dimension in the ambient space, while the boundary of the Mandelbrot set has dimension 2.
Also it is not quite true that all Julia sets are fixed sets for Hutchinson operators, at least under the usual definition of Hutchinson operator which requires strict contractivity of the maps (the maps $z\to\pm\sqrt{z-c}$ are in general not contractions, even on the Julia set itself)
