This study presents an analysis of the dynamics of single and multiple chains of spherical super-paramagnetic beads suspended in a Newtonian fluid under the combined effect of an external rotating magnetic field and a shear flow. Viscosity results depend on two main non-dimensional numbers: the ratio between the shear rate and the magnetic rotation frequency and the ratio between the hydrodynamic and magnetostatic interactions (the Mason number). When the shear rate is smaller than the magnetic field frequency, the chain rotation accelerates the surrounding fluid, reducing the value of the measured suspension viscosity even below that of the solvent. In this regime, shear-thickening is observed. For values of the shear rates comparable to the rotation magnetic frequency, the viscosity reaches a maximum and non-linear coupling effects come up. If the shear rate is increased to values above the rotation frequency, the viscosity decreases and a mild shear-thinning is observed. In terms of the Mason number, the suspension viscosity reduces in line with the literature results reported for fixed magnetic fields, whereas the shear-rate/magnetic-frequency ratio parameters induce a shift of the viscosity curve towards larger values. Results at larger concentrations and multiple chains amplify the observed effects.