Holomorphic motions provide a bridge between analysis and complex dynamics, and provide also powerful tools for the quasiconformal mappings. Basic examples of holomorphic motions are given by Julia sets when parameters of polynomials are varied, and these give important examples within different topics in analysis, as well. In this talk, based on a joint work with Ivrii, Prause and Perälä, we are interested in maximal growth of dimension under holomorphic motions of Julia sets. In particular, consider the family $P(z) = z^d + tz$ with $|t| <1$. Slodkowski's theorem allows for the Böttcher coordinates a natural extension to a holomorphic motion of the plane. But can there exist a better one ?