Reproducible low bias bipolar resistive switching memory in HfZnOx based memristors is reported. The modification of the concentration of oxygen vacancies in the ternary oxide film, which is facilitated by adding ZnO into HfO2, results in improved memory operation by the ternary oxide compared to the single binary oxides. A controlled multistate memory operation is achieved by controlling current compliance and RESET stop voltages. A high DC cyclic stability up to 400 cycles in the multistate memory performance is observed. Conventional synaptic operation in terms of potentiation, depression plasticity, and Ebbinghaus forgetting process are also studied. The memory mechanism is shown to originate from the migration of the oxygen vacancies and modulation of the interfacial layers.