The connectedness locus is a generalization of the classical Mandelbrot set in the case of higher degree polynomials with one critical point. I will discuss some contributions to the old problem of describing the similarity between a neighborhood of a point on the boundary of the connectedness locus and the corresponding Julia set. There clearly are limitations on such similarity, so the exact statements have to be careful. Loosely speaking, however, for a typical point in the sense of the harmonic measure on the boundary of the connectedness locus we manage to construct a similarity map which is quasiconformal on a ball and conformal, i.e. complex-differentiable with non-zero derivative, at the point itself.