  We investigate a generalized Hadamard walk in two dimensions with five inner
states. The particle governed by a five-state quantum walk (5QW) moves, in
superposition, either leftward, rightward, upward, or downward according to the
inner state. In addition to the four degrees of freedom, it is allowed to stay
at the same position. We calculate rigorously the wave function of the particle
starting from the origin in the plane for any initial state, and give the
spatial distribution of probability of finding the particle. We also
investigate the localization problem for the two-dimensional five-state quantum
walk: Does the probability of finding a particle anywhere on the plane converge
to zero even after infinite time steps except initial states?
