Approximating the chromatic polynomial is as hard as computing it exactly

We show that for any non-real algebraic number $q$ such that $\mathrm{\mid }q-1\mathrm{\mid }>1$ or $\mathrm{\Re }(q)>\frac{3}{2}$ it is $\text{textsc}\mathrm{\#}P$-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 $\text{textsc}\mathrm{\#}P$-hardness for all non-real algebraic $q$ on the family of all graphs. We moreover prove several hardness results for $q$ such that $\mathrm{\mid }q-1\mathrm{\mid }\le 1$. 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.