  We define a quaternionic extension of the Szegedy walk on a graph and study
its right spectral properties. The condition for the transition matrix of the
quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In
order to derive the spectral mapping theorem for the quaternionic Szegedy walk,
we derive a quaternionic extension of the determinant expression of the second
weighted zeta function of a graph. Our main results determine explicitly all
the right eigenvalues of the quaternionic Szegedy walk by using complex right
eigenvalues of the corresponding doubly weighted matrix. We also show the way
to obtain eigenvectors corresponding to right eigenvalues derived from those of
doubly weighted matrix.
