  In recent years there has been a resurgence of interest in Bohmian mechanics
as a numerical tool because of its local dynamics, which suggest the
possibility of significant computational advantages for the simulation of large
quantum systems. However, closer inspection of the Bohmian formulation reveals
that the nonlocality of quantum mechanics has not disappeared -- it has simply
been swept under the rug into the quantum force. In this paper we present a new
formulation of Bohmian mechanics in which the quantum action, S, is taken to be
complex. This leads to a single equation for complex S, and ultimately complex
x and p but there is a reward for this complexification -- a significantly
higher degree of localization. The quantum force in the new approach vanishes
for Gaussian wavepacket dynamics, and its effect on barrier tunneling processes
is orders of magnitude lower than that of the classical force. We demonstrate
tunneling probabilities that are in virtually perfect agreement with the exact
quantum mechanics down to 10^{-7} calculated from strictly localized quantum
trajectories that do not communicate with their neighbors. The new formulation
may have significant implications for fundamental quantum mechanics, ranging
from the interpretation of non-locality to measures of quantum complexity.
