Despite being known of for decades, the actual realization of memory devices based on the memristive effect is progressing slowly, due to processing requirements and the need for exotic materials which are not compatible with today's complementary-metal-oxide-semiconductor (CMOS) technology. Here, we report an experimental study on a Ge quantum wire device featuring distinct signatures of memristive behavior favorable for integration in CMOS platform technology. Embedding the quasi-1D Ge quantum wire into an electrostatically modulated back-gated field-effect transistor, we demonstrate that individual current transport channels can be addressed directly by controlling the surface trap assisted electrostatic gating. The resulting quantization of the current represents the ultimate limit of memristors with practically zero off-state current and low footprint. In addition, the proposed device has the advantage of non-destructive successive reading cycles capability. Importantly, our findings provide a framework towards fully CMOS compatible ultra-scaled Ge based memristors.