Oxide heterostructures are versatile platforms with which to research and create novel functional nanostructures. We successfully develop one-dimensional (1D) quantum-wire devices using quantum point contacts on MgZnO/ZnO heterostructures and observe ballistic electron transport with conductance quantised in units of 2e^2/h. Using DC-bias and in-plane field measurements, we find that the g-factor is enhanced to around 6.8, more than three times the value in bulk ZnO. We show that the effective mass m^* increases as the electron density decreases, resulting from the strong electron-electron interactions. In this strongly interacting 1D system we study features matching the '0.7' conductance anomalies up to the fifth subband.(up to ~5.7 x 2e^2/h). This paper demonstrates that high-mobility oxide heterostructures such as this can provide good alternatives to conventional III-V semiconductors in spintronics and quantum computing as they do not have their unavoidable dephasing from nuclear spins. This paves a way for the development of qubits benefiting from the low defects of an undoped heterostructure together with the long spin lifetimes achievable in silicon.