This paper presents our achievement in the development of new types of advanced composite scintillators (CSs) based on the single crystalline films (SCFs) of Lu3Al5O12 (LuAG) garnet, doped with Pr3+ and Sc3+ ions, and substrate from single crystals (SCs) of Ce3+-doped LuAG using the technology of liquid phase epitaxy. We show the possibility of the simultaneous registration of $\alpha$ -particles and $\gamma$ -quanta by means of separation of the decay kinetics of SCF and crystal parts of such CSs. Namely, the significant differences in the scintillation decay kinetics of LuAG:Pr SCF/LuAG:Ce SC and LuAG:Sc SCF/LuAG:Ce SC CSs under excitation by $\alpha$ -particles of 241Am (5.5 MeV) source and $\gamma$ -quanta of 137Cs (662 keV) source are observed. Thus, such types of CSs can be successfully applied for separation of the signals coming from their film and crystal parts at the registration of the mixed radiation fluxes of $\alpha$ -particles and $\gamma$ -quanta.