  We define the quaternionic quantum walk on a finite graph and investigate its
properties. This walk can be considered as a natural quaternionic extension of
the Grover walk on a graph. We explain the way to obtain all the right
eigenvalues of a quaternionic matrix and a notable property derived from the
unitarity condition for the quaternionic quantum walk. Our main results
determine all the right eigenvalues of the quaternionic quantum walk by using
complex eigenvalues of the quaternionic weighted matrix which is easily
derivable from the walk. Since our derivation is owing to a quaternionic
generalization of the determinant expression of the second weighted zeta
function, we explain the second weighted zeta function and the relationship
between the walk and the second weighted zeta function.
