  We show that the semi-classical analysis of generic Euclidean path integrals
necessarily requires complexification of the action and measure, and
consideration of complex saddle solutions. We demonstrate that complex saddle
points have a natural interpretation in terms of the Picard-Lefschetz theory.
Motivated in part by the semi-classical expansion of QCD with adjoint matter on
${\mathbb R}^3\times S^1$, we study quantum-mechanical systems with bosonic and
fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as
well as (related) purely bosonic systems with harmonic non-degenerate minima.
We find exact finite action non-BPS bounce and bion solutions to the
holomorphic Newton equations. We find not only real solutions, but also complex
solution with non-trivial monodromy, and finally complex multi-valued and
singular solutions. Complex bions are necessary for obtaining the correct
non-perturbative structure of these models. In the supersymmetric limit the
complex solutions govern the ground state properties, and their contribution to
the semiclassical expansion is necessary to obtain consistency with the
supersymmetry algebra. The multi-valuedness of the action is either related to
the hidden topological angle or to the resurgent cancellation of ambiguities.
We also show that in the approximate multi-instanton description the
integration over the complex quasi-zero mode thimble produces the most salient
features of the exact solutions. While exact complex saddles are more difficult
to construct in quantum field theory, the relation to the approximate thimble
construction suggests that such solutions may be underlying some remarkable
features of approximate bion saddles in quantum field theories.
