  We extend the theory of Kapitza stabilization within the complex domain, i.e.
for the case of an imaginary oscillating potential. At a high oscillation
frequency, the quasi-energy spectrum is found to be entirely real-valued,
however a substantial difference with respect to a real potential emerges, that
is the formation of a truly bound state instead of a resonance. The predictions
of the Kapitza averaging method and the transition from a complex to an
entirely real-valued quasi-energy spectrum at high frequencies are confirmed by
numerical simulations of the Schrodinger equation for an oscillating Gaussian
potential. An application and a physical implementation of the imaginary
Kapitza pendulum to the stability of optical resonators with variable
reflectivity is discussed.
