We present the combination of microscopic model calculations and direct experimental evidence for complex spatio-temporal dynamics on a picosecond timescale in the near-field of a broad-area semiconductor laser induced by the interplay of self-focusing, diffraction, and spatial hole burning. We demonstrate the formation of migrating optical filaments within the turn-on dynamics of such a laser and characterize the resulting spatio-temporal complexity via spatio-temporal cross-correlation functions.