This article presents the generalization of the unsteady MHD free convection flow of non-Newtonian sodium alginate-ferrimagnetic nanofluid in two infinite vertical parallel plates. The different shape (blade, brick, cylinder, and platelet) ferrimagnetic nanoparticles are dissolved in the non-Newtonian sodium alginate (SA) as base fluid to form non-Newtonian nanofluids. The Jeffrey fluid model together with energy equation is considered to demonstrate the flow. The Atangana–Baleanu fractional operator is utilized for the generalization of mathematical model. The Laplace transform technique and Zakian's numerical algorithm are used to developed general solutions with a fractional order for the proposed model. The obtained results are computed numerically and presented graphically to understand the physics of pertinent flow parameters. It is noticed that the velocity and temperature profiles are significantly increased with the increasing values of the fractional parameter due to the variation in thermal and momentum boundary layers. In the case of the effect of different shapes of nanoparticles, density is a dominant factor as compared to thermal conductivity, which significantly affects the flow of non-Newtonian nanofluid.