This paper reports a numerical study of complex classical trajectories of a particle in an elliptic potential. This study of doubly periodic potentials is a natural sequel to earlier work on complex classical trajectories in trigonometric potentials. For elliptic potentials, there is a two-dimensional array of identical cells in the complex plane, and each cell contains a pair of turning points. The particle can travel both horizontally and vertically as it visits these cells, and sometimes the particle is captured temporarily by a pair of turning points. If the particle's energy lies in a conduction band, the particle drifts through the lattice of cells and is never captured by the same pair of turning points more than once. However, if the energy of the particle is not in a conduction band, the particle can return to previously visited cells.