The processes of jetting and drop formation is strongly affected by fluid rheology, which may be complex, particularly under the extreme conditions of high shear and extensions rates that occur during jetting. Fluids containing a particulate phase are normally shear-thinning and so may have different characteristic viscosities during different key stages of the inkjet flow. Moreover, even trace amounts of long chain polymers can cause substantially different breakup dynamics compared to that of an ordinary (Newtonian) fluid. In this work we investigate the dependency of jet breakup behaviour upon viscoelastic and shear-thinning effects in the context of drop-on-demand inkjet drop formation. In drop-on-demand printing, each ejected drop remains connected temporarily to the printhead by a trailing ligament of fluid which undergoes capillary thinning while the drop is in flight. Upon pinch-off the severed ligament may recoil downstream towards the leading drop, or alternatively it may fragment into multiple satellite droplets. Whilst complex rheology is often seen as a problem, particularly given the lack of instrumentation able to measure and characterize fluid properties at the appropriate deformation rates and timescales, it also offers a potential solution to controlling satellite drops at higher printing speeds. We show the results of numerical simulations of drop-on demand inkjet printing with fluids that exhibit different types of non-Newtonian behaviour (shear-thinning and viscoelasticity) and compare with experiments on model inks. Our aim is to establish the parameter values controlling the break-up length and character of jet break-up. In particular, we examine whether for appropriate choices of rheological parameters it is possible to prevent or impede the creation of satellite drops without compromising on printing speed.