  We establish the quaternionic weighted zeta function of a graph and its Study
determinant expressions. For a graph with quaternionic weights on arcs, we
define a zeta function by using an infinite product which is regarded as the
Euler product. This is a quaternionic extension of the square of the Ihara zeta
function. We show that the new zeta function can be expressed as the
exponential of a generating function and that it has two Study determinant
expressions, which are crucial for the theory of zeta functions of graphs.
