We show how Feynman's path integral for quantum mechanics may be defined without a Wick rotation to imaginary time. Instead, we employ analytic continuation (and Cauchy's theorem) in the complexified space of paths being integrated over. We outline an existence proof and describe applications to both nonrelativistic quantum mechanics and to interference patterns due to gravitational microlensing in radio astronomy. [please email a.held@imperial.ac.uk for zoom link or password]