  Recent advances in cold-atom platforms have made real-time dynamics
accessible, renewing interest in the motion of superfluid vortices in
two-dimensional domains. Here we show that the energy and the trajectories of
arbitrary vortex configurations may be computed on a complicated (curved or
bounded) surface, provided that one knows a conformal map that links the latter
to a simpler domain (like the full plane, or a circular boundary). We also
prove that Hamilton's equations based on the vortex energy agree with the
complex dynamical equations for the vortex dynamics, demonstrating that the
vortex trajectories are constant-energy curves. We use these ideas to study the
dynamics of vortices in a two-dimensional incompressible superfluid with an
elliptical boundary, and we derive an analytical expression for the complex
potential describing the hydrodynamic flow throughout the fluid. For a vortex
inside an elliptical boundary, the orbits are nearly self-similar ellipses.
