Magnetic microparticles suspended on the surface of liquid and energized by vertical alternating magnetic field exhibit complex collective behavior. Various immobile and self-propelled self-assembled structures have been observed. Here, we report on experimental studies of mixing and surface diffusion processes in this system. We show that the pattern-induced surface flows have properties of quasi-two-dimensional turbulence. Correspondingly, the surface advection of tracer particle exhibits properties of Brownian diffusion.