  We study the complexity of approximating the partition function of the
$q$-state Potts model and the closely related Tutte polynomial for complex
values of the underlying parameters. Apart from the classical connections with
quantum computing and phase transitions in statistical physics, recent work in
approximate counting has shown that the behaviour in the complex plane, and
more precisely the location of zeros, is strongly connected with the complexity
of the approximation problem, even for positive real-valued parameters.
Previous work in the complex plane by Goldberg and Guo focused on $q=2$, which
corresponds to the case of the Ising model; for $q>2$, the behaviour in the
complex plane is not as well understood and most work applies only to the
real-valued Tutte plane.
  Our main result is a complete classification of the complexity of the
approximation problems for all non-real values of the parameters, by
establishing \#P-hardness results that apply even when restricted to planar
graphs. Our techniques apply to all $q\geq 2$ and further complement/refine
previous results both for the Ising model and the Tutte plane, answering in
particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in
the context of quantum computations.
