The complex quantum Hamilton-Jacobi equation (CQHJE) for the action function is integrated by propagating correlated Bohmian trajectories in real space. We transform the CQHJE into the arbitrary Lagrangian-Eulerian (ALE) version with the grid velocity matching the flow velocity of the probability fluid. The ALE version describes the rate of change in the complex action transported along Bohmian trajectories. The spatial derivatives of the complex action required for the integration of the CQHJE are evaluated with a moving least squares algorithm. The method is applied to a squeezed state in the harmonic potential and to a two-dimensional barrier scattering problem. (C) 2013 Elsevier B.V. All rights reserved.