  We discuss path integrals for quantum mechanics with a potential which is a
perturbation of the upside-down oscillator. We express the path integral (in
the real time) by the Wiener measure. We obtain the Feynman integral for
perturbations which are the Fourier-Laplace transforms of a complex measure and
for polynomials of the fotm $x^{4n}$ and $x^{4n+2}$ (where $n$ is a natural
number). We extend the method to quantum field theory (QFT) with complex scaled
spatial coordinates ${\bf x}\rightarrow i{\bf x}$. We show that such a complex
extension of the path integral (in the real time) allows a rigorous path
integral treatment of a large class of potentials including the ones unbounded
from below.
