Abstract: We consider a flat complex vector bundle over a compact Riemannian manifold with boundary. We suppose it admits a fiberwise non-degenerate symmetric complex bilinear form. We use the Riemannian metric on the manifold and this bilinear form to define non self-adjoint Laplacians acting on (vector valued) differential forms, which satisfy relative (resp. absolute) boundary conditions. With these conditions, this is an elliptic boundary problem. By using some regularity theory and general properties of closed operators on Banach spaces, we obtain a Hodge decomposition result for these Laplacians. As a consequence, we show that their (generalized) kernels compute relative (resp. absolute) de Rham cohomology groups.