In this paper we consider the solution of the enhanced Boussinesq equations of Madsen and S$\oo$rensen ({\it Coast.Eng.} 18, 1992) by means of residual based discretizations. In particular, we investigate the applicability of upwind and stabilized variants of the Residual Distribution and Galerkin finite element schemes for the simulation of wave propagation and transformation over complex bathymetries. These techniques have been successfully applied to the solution of the nonlinear Shallow Water equations (Ricchiuto and Bollerman {\it J.Comput.Phys} 228, 2009 - Hauke {\it CMAME} 163, 1998). The work discussed in this paper constitutes a first step toward the obtention of a model coupling the enhanced Boussinesq equations with the Shallow Water equations in wave breaking regions. The contribution of the present work is to show that equal order and even low order (second) upwind/stabilized techniques can be used to model non-hydrostatic wave propagation over complex bathymetries. This result is supported by theoretical (truncation and dispersion) error analyses, and by a thorough numerical validation.