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Half topological states of matter and two-dimensional (2D) magnetism have gained much attention recently. In this paper, we propose a special topological semimetal phase called a 2D half Chern-Weyl semimetal (HCWS), which is a 2D magnetic Weyl semimetal bound to the half Chern insulator phase by symmetry, and the two phases can be converted to each other by manipulating the magnetization direction. We provide the symmetry conditions to realize this state in 2D systems with multiple screw axes. Tight-binding models with multiple basis and a predicted 2D material, monolayer TiTe, are shown as the concrete examples for HCWSs. The TiTe monolayer was shown to have a high ferremagnetic Curie temperature (~966 K) as well as a Coulomb correlation-enhanced spin-orbit coupling (SOC), and further demonstrates the effect of correlation-enhanced SOC on magnetocrystalline anisotropy energy and energy gap opening. Our work reveals a state with switchable and spin-resolved half body charge currents as well as half boundary charge currents, and will provide a platform for novel and high-performance topological spintronics devices.