  Generic higher derivative theories are believed to be fundamentally
unphysical because they contain Ostrogradsky ghosts. We show that within
complex classical mechanics it is possible to construct higher derivative
theories that circumvent the Ostrogradsky theorem and have a real energy
spectrum that is bounded from below. The complex theory can be canonically
quantised. The resulting quantum theory does not suffer from the kinetic
instability and maintains the usual probabilistic interpretation without
violating the correspondence principle. As a proof of concept, we construct a
class of stable interacting complex higher derivative theories. This consistent
and canonical framework allows us to analyse previous attempts to avoid the
ghosts that use non-canonical quantisation schemes, such as the Lee-Wick
theories, Dirac-Pauli quantisation or PT-symmetric quantum mechanics. The key
to understand the would-be ghosts in any kinetically stable higher derivative
theory is to accept the complex system behind it.
