Self-mating of the golden mean Siegel quadratic polynomial, courtesy of J. Milnor.
One can get a 2-sphere by sewing two copies of this extremely wild fractal (one in
blue and the other in yellow) along the boundary in a combinatorially precise way.
See Mating Siegel Quadratic Polynomials, J. Amer. Math. Soc. 14 (2001) 25-78.