Abstract: A two-body fully relativistic three-dimensional scattering framework has been utilised to develop an alternative Compton scattering computational model to those adapted from Ribberfors' work for Monte Carlo modelling of Compton scattering. Using a theoretical foundation that ensures the conservation of energy and momentum in the relativistic impulse approximation, this new model, the Monash University Compton scattering model, develops energy and directional algorithms for both the scattered photon and ejected Compton electron from first principles. The Monash University Compton scattering model was developed to address the limitation of the Compton electron directionality algorithms of other computational models adapted from Ribberfors' work. Here the development of the Monash University Compton scattering model, including its implementation in a Geant4 low energy electromagnetic physics class, G4LowEPComptonModel, is outlined. Assessment of the performance of G4LowEPComptonModel was undertaken in two steps: (1) comparison with respect to the two standard Compton scattering classes of Geant4 version 9.5, G4LivermoreComptonModel and G4PenelopeComptonModel, and (2) experimental comparison with respect to Compton electron kinetic energy spectra obtained from the Compton scattering of 662 key photons off the K-shell of gold. Both studies illustrate that the Monash University Compton scattering model, and in turn G4LowEPComptonModel, is a viable replacement for the majority of computational models that have been adapted from Ribberfors' work. It was also shown that the Monash University Compton scattering model is able to reproduce the Compton scattering triply differential cross-section Compton electron kinetic energy spectra of 662 keV photons K-shell scattering off of gold to within experimental uncertainty.