We study an analytically solvable and experimentally relevant number-conserving periodically driven (p)-wave superconductor. Such a system is found to support generalized Majorana zero and (\pi) modes which, despite being non-Hermitian, are still capable of encoding qubits. Moreover, appropriate winding numbers characterizing the topology of such generalized Majorana modes are defined and explicitly calculated. We further discuss the fate of the obtained generalized Majorana modes in the presence of finite charging energy. Finally, we shed light on the quantum computing prospects of such modes by demonstrating the robustness of their encoded qubits and explicitly braiding a pair of generalized Majorana modes.