Informally speaking, a compact subset of a plane is called computable if there is an algorithm which can draw arbitrarily good approximations of this set. Computational complexity measures how long does it take to draw these approximations. It is known that for some classes of rational maps (e.g. hyperbolic, parabolic and Collet-Eckmann) the Julia sets have polynomial complexity. For others (e.g. Siegel) the Julia sets can have arbitrarily high computational complexity and even may be uncomputable. However, not much is known in the case of presence of Cremer periodic points. We show that there exist abundant Cremer quadratic polynomials with Julia sets of arbitrarily high complexity. The talk is based on a joint work with Michael Yampolsky.