The spiky empanada

This is a picture of the basin of attraction for the origin under the map (z,w) |-> (z^2-3*w^2,w^2), rendered in three dimensions by means of the Hopf fibration. The smooth part of the boundary is foliated by Riemann surfaces. Bounded holomorphic functions on such a surface extend (in some sense) to the basin, so that the basin embeds into the corona of the surface. For details see the papers A new construction of Riemann surfaces with corona (with Jeff Diller) and Holomorphic extension from boundaries with concentrated Levi form.