  We investigate the cosmological model with the complex scalar
self-interacting inflaton field non-minimally coupled to gravity. The different
geometries of the Euclidean classically forbidden regions are represented. The
instanton solutions of the corresponding Euclidean equations of motion are
found by numerical calculations supplemented by the qualitative analysis of
Lorentzian and Euclidean trajectories. The applications of these solutions to
the no-boundary and tunneling proposals for the wave function of the Universe
are studied. Possible interpretation of obtained results and their connection
with inflationary cosmology is discussed. The restrictions on the possible
values of the new quasi-fundamental constant of the theory-non-zero classical
charge-- are obtained. The equations of motion for the generalized cosmological
model with complex scalar field are written down and investigated. The
conditions of the existence of instanton solutions corresponding to permanent
values of an absolute value of scalar field are obtained.
