Instead of a plane, if we had an image on a sphere $x^2+y^2+z^2=1$ and wanted to conformally map this into another image on a sphere. What would the equivalent Cauchy-Riemann equations be for this?
i.e. we would have the coordinates of the new image $(x',y',z')$ such that:
$x' = f(x,y,z), y'=g(x,y,z), z'=h(x,y,z)$.
and $f^2+g^2+h^2=1$ and all angles preserved. I presume one could form some differential equation.