Properties of iron oxides at the extreme conditions are of essential importance in condensed matter physics and Geophysics. The recent discovery of a new type of iron oxide, Fe4O5, at high pressure and high temperature of Earths deep interior attracts great interests. In this paper, we report the electronic structure and the magnetic properties of Fe4O5 predicted by the density functional theory plus dynamic mean field theory (DFT+DMFT) approach. We find that Fe4O5 stays metallic from ambient pressure to high pressure. The magnetic moments of iron atoms at the three different crystallographic positions of Fe4O5 undergo position-dependent collapse as being compressed. Such site-selective magnetic moment collapse originates from the shift of energy levels and the consequent charge transfer among the Fe-3d orbits under compression.