Abstract: Yves Couder and coworkers have recently reported the results of a startling series of experiments in which droplets walking on a vibrating fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm, including single- particle diffraction and interference, tunneling and quantized orbits. A correspondence is developed between this fluid system and de Broglie’s pilot wave theory of relativistic quantum mechanics. In the shallow water limit of the fluid system, one can infer hydrodynamic analogs of Planck’s constant, the speed of light, the Compton and de Broglie wavelengths. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.