The possible occurence of highly deformed configurations in the $^{40}$Ca di-nuclear system formed in the $^{28}$Si + $^{12}$C reaction is investigated by analyzing the spectra of emitted light charged particles. Both inclusive and exclusive measurements of the heavy fragments (A $geq$ 10) and their associated light charged particles (protons and $alpha$ particles) have been made at the IReS Strasbourg {sc VIVITRON} Tandem facility at bombarding energies of $E_{lab}$ ($^{28}$Si) = 112 MeV and 180 MeV by using the {sc ICARE} charged particle multidetector array. The energy spectra, velocity distributions, in-plane and out-of-plane angular correlations of light charged particles are compared to statistical-model calculations using a consistent set of parameters with spin-dependent level densities. This spin dependence approach suggests the onset of large nuclear deformation in $^{40}$Ca at high spin. This conclusion might be connected with the recent observation of superdeformed bands in the $^{40}$Ca nucleus. The analysis of $alpha$ particles in coincidence with $^{32}$S fragments suggests a surprisingly strong $^{8}$Be cluster emission of a binary nature.