We collect some recent results that together provide an almost complete answer to the question stated in the title. For the dimension d=2 the answer is three. For the dimensions d=3 and d>4 the answer is four. For the dimension d=4 the answer is either three or four. Curiously, the exact number in d=4 seems to be an open problem.
@article{arxiv.1504.01590,
title = {How many orthonormal bases are needed to distinguish all pure quantum states?},
author = {Claudio Carmeli and Teiko Heinosaari and Jussi Schultz and Alessandro Toigo},
journal= {arXiv preprint arXiv:1504.01590},
year = {2015}
}