In this Letter we report a novel self-organized copper electrodeposition in an ultrathin layer of CuSO4 electrolyte. The macroscopic fingering branches of the deposit consist of long copper filaments covered with periodic corrugated nanostructures. The mechanism of the nanostructure formation is explored and the origin of the significant descent of the branching rate in electrodeposition is discussed. We suggest that this growth phenomenon provides deeper insights into the role of diffusion and migration on pattern formation in electrodeposition.