In this paper we consider the solution of the enhanced Boussinesq equations of Madsen and Sorensen (1992) [55] by means of residual based discretizations. In particular, we investigate the applicability of upwind and stabilized variants of continuous Galerkin finite element and Residual Distribution 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 (see e.g. Hauke (1998) [39] and Ricchiuto and Bollermann (2009) [61]). In a first step toward the construction of a hybrid model coupling the enhanced Boussinesq equations with the Shallow Water equations in breaking regions, this paper shows 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 thorough numerical validation.