We present and analyze Schwarz domain decomposition methods for a general diffusion problem with complex advection. The complex advection term changes completely the nature of the solution and makes it more Helmholtz like. We analyze in detail the influence of the outer boundary conditions on the performance of the Schwarz algorithm, including PML conditions to emulate free space problems, and optimized transmission conditions, also for multiple subdomains. Our results show that the performance of Schwarz methods for such Helmholtz like problems is much better on free space configurations than in waveguides or closed cavities. Equations with complex advection appear in diverse applications, for example the convected Helmholtz equation, the Gross-Pitaevskii equation, Schrödinger equations, and also as important component in the wave-ray multigrid algorithm for Helmholtz problems. We show as an example the performance of our Schwarz methods for a potential flow around a schematic submarine.