The talk concerns the ongoing development of a non-standard model of continuum mechanics, originally due to Godunov, Peshkov, and Romenski (GPR), and its numerical approximation in Finite Volume and Discontinuous Galerkin methods. The main feature of the model is that it describes a general continuum, rather than a classic fluid or solid medium, with the difference between the two being specified only by a choice of parameters. In this framework, rather general closure laws can be implemented, including non-Newtonian rheologies, visco-elasto-plasticity, material damage and fractures, melting and solidification, and more. The model is cast in a first order hyperbolic form with stiff relaxation sources, which means that it requires no second order diffusive fluxes, and that it yields a theory in which all signals propagate with finite speed, including heat conduction. A clear drawback of the model is its complexity, in particular when applied to Newtonian viscous fluids and compared to the well established Navier-Stokes equations. Together with stiff sources, one has to also consider the presence of differential involutions and algebraic constraints, together with other nonlinearities and representation issues concerning the evolution of matrix-valued data. Here I outline my efforts towards closing the complexity gap and making the formalism more accessible, mainly focusing on the treatment of stiff sources, algebraic constraints, and on new resolution improvements involving the formulation and solution of a quaternion-valued PDE.