In the past few years, there has been a notable global surge in research on nanofluids, driven by their promising thermal applications in engineering and biological sciences. Nanofluids have demonstrated promising results in enhancing heat transfer phenomena. To further enhance the thermal performance of conventional base fluids, researchers have increasingly focused on investigating the use of structured nanoparticle suspensions within these fluids. With a consideration of the potential applications of nanoparticles, this paper intends to explore the utilization of three nanoparticles with distinct shapes within a single base fluid. More precisely, three different nanoparticles with different shapes, i.e., spherical-shaped gold (Au), cylindrical-shaped zinc (Zn), and platelet-shaped ferric oxide (Fe3O4) are added to the base fluid blood because of their relative advanced pharmaceutical applications. In this study, the primary focus is to thoroughly analyze the heat transfer characteristics of an unsteady flow of a couple-stress Casson ternary hybrid nanofluid within a channel. The flow regime under investigation is represented by classical partial differential equations, which are subsequently non-dimensionalized using appropriate non-dimensional variables. To further analyze the system, the dimensionless partial differential equations are fractionally modified using Caputo's definition of fractional derivatives, incorporating Fick's and Fourier's laws, and the exact solutions for temperature, concentration, and velocity profiles are achieved by employing the Laplace and Fourier transforms. The results clearly indicate that as the volume fraction of nanoparticles increases, the fluid velocity decreases while the temperature rises. The utilization of a blood-based ternary hybrid nanofluid enhances the rate of heat transfer by up to 20%. Specifically, the inclusion of spherical-shaped gold (Au) nanoparticles rises heat transfer by up to 16%, cylindrical-shaped zinc (Zn) nanoparticles enhance it by up to 19%, and platelet-shaped ferric oxide (Fe3O4) nanoparticles enhance it by up to 23%.