Boron phosphide has recently been identified as a potential high-hole-mobility transparent conducting material. This promise arises from its low hole effective masses. However, boron phosphide has a relatively small, 2 eV, indirect bandgap which will affect its transparency. In this work, we computationally study both optical absorption across the indirect gap and phonon-limited electronic transport to quantify the potential of boron phosphide as a p-type transparent conductor. We find that phonon-mediated indirect optical absorption is weak in the visible spectrum and that the phonon-limited hole mobility is very high (around 900 cm2/Vs) at room temperature. This exceptional mobility comes from the combination of a low hole effective mass and very weak scattering by polar phonon modes. We rationalize the weak scattering by the less ionic bonding in boron phosphide compared to oxides. We suggest that this could be a general advantage of nonoxides for p-type transparent conducting applications. Using our computed properties, we assess the transparent conductor figure of merit of boron phosphide and show that it exceeds by one order of magnitude that of established p-type transparent conductors, confirming the potential of this material.