Atomically thin two-dimensional (2D) materials have received considerable research interest due to their extraordinary properties and promising applications. Here we predict the monolayered indium triphosphide (InP3) as a new semiconducting 2D material with a range of favorable functional properties by means of ab initio calculations. The 2D InP3 crystal shows high stability and promise of experimental synthesis. It possesses an indirect band gap of 1.14 eV and a high electron mobility of 1919 cm2 V-1 s-1, which can be strongly manipulated with applied strain. Remarkably, the InP3 monolayer suggests tunable magnetism and half-metallicity under hole doping or defect engineering, which is attributed to the novel Mexican-hat-like bands and van Hove singularities in its electronic structure. A semiconductor-metal transition is also revealed by doping 2D InP3 with electrons. Furthermore, monolayered InP3 exhibits extraordinary optical absorption with significant excitonic effects in the entire range of the visible light spectrum. All these desired properties render 2D InP3 a promising candidate for future applications in a wide variety of technologies, in particular for electronic, spintronic, and photovoltaic devices.