We propose and demonstrate numerically a measurement scheme for complete reconstruction of the 2D quantum wave function of a Bose-Einstein condensate, amplitude and phase, from a time-of-flight measurement. We identify a fundamental ambiguity present in the measurement of phase structures of high-symmetry excitations (e.g., vortices) and show how to overcome it by allowing for different expansion durations in different directions. We demonstrate this approach with the reconstruction of matter-wave vortices and arrays of vortices.