The present study investigates magnetohydrodynamic (MHD) bioconvective flow and transport characteristics of a Carreau hybrid nanofluid (HNF) over a nonlinear stretching surface embedded in a porous medium. The hybrid nanoparticle suspension (Fe3O4 + CoFe2O4 in water) accounts for thermal radiation, activation energy, Brownian motion, thermophoresis, and gyrotactic microorganisms. By using similarity variables, the original PDEs describing the process are converted and then solved numerically through an adaptive Runge–Kutta-Fehlberg (RKF-45) shooting technique. Findings show that hybrid nanoparticles can be used to increase the heat transfer rate up to 31% and mass transfer approximately by 23%. The magnetic parameter acts to reduce the flow velocity due to Lorentz force, additionally, the radiation parameter and Eckert number upsurge temperature. The Peclet number decreases the distribution of microorganisms while the bioconvective Lewis number promotes their concentration in the region. Dilatant fluids give rise to stronger heat transfer whereas in pseudoplastic fluids their mass diffusion is facilitated more. The results complement well developing thermal management and energy utilization systems.