Quantum vortices are the principal excitations in many macroscopic quantum systems, including superfluids and cold atom Bose-Einstein condensates. They are characterized by a quantized phase circulation of the wave function around a vortex core. Here we employ the Gross-Pitaevskii equation to investigate the structure of minimum energy vortices in cold atom Bose-Einstein condensates, in a nonrotating axially symmetric harmonic trap. For a generic value of the angular momentum along the symmetry axis, the energy minima are eccentric vortices. We find that the vortices precess around the center of the trap uniformly, in a timecrystalline fashion. Furthermore, we demonstrate that when two identical vortices exchange their position, the wave function acquires a phase with an anyonic character. Our results reveal that quantum vortices have an unexpectedly rich phenomenology, suggestive of applications to emerging subjects from quantum computation to simulation and information processing.