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.