  We show that for any non-real algebraic number $q$ such that $|q-1|>1$ or
$\Re(q)>\frac{3}{2}$ it is \textsc{\#P}-hard to compute a multiplicative (resp.
additive) approximation to the absolute value (resp. argument) of the chromatic
polynomial evaluated at $q$ on planar graphs. This implies
\textsc{\#P}-hardness for all non-real algebraic $q$ on the family of all
graphs. We moreover prove several hardness results for $q$ such that $|q-1|\leq
1$.
  Our hardness results are obtained by showing that a polynomial time algorithm
for approximately computing the chromatic polynomial of a planar graph at
non-real algebraic $q$ (satisfying some properties) leads to a polynomial time
algorithm for \emph{exactly} computing it, which is known to be hard by a
result of Vertigan. Many of our results extend in fact to the more general
partition function of the random cluster model, a well known reparametrization
of the Tutte polynomial.
