We investigate the cosmological model with the complex scaler self-interact
ing inflaton field non-minimally coupled to gravity. The different geometri
es of the Euclidean classically forbidden regions are represented. The inst
anton solutions of the corresponding Euclidean equations of motion are foun
d by numerical calculations supplemented by the qualitative analysis of Lor
entzian and Euclidean trajectories. The applications of these solutions to
the no-boundary and tunneling proposals for the wave function of the Univer
se are studied. Possible interpretation of obtained results and their conne
ction with inflationary cosmology is discussed.