  We study the complexity of approximating the partition function
$Z_{\mathrm{Ising}}(G; \beta)$ of the Ising model in terms of the relation
between the edge interaction $\beta$ and a parameter $\Delta$ which is an upper
bound on the maximum degree of the input graph $G$. Following recent trends in
both statistical physics and algorithmic research, we allow the edge
interaction $\beta$ to be any complex number. Many recent partition function
results focus on complex parameters, both because of physical relevance and
because of the key role of the complex case in delineating the
tractability/intractability phase transition of the approximation problem. In
this work we establish both new tractability results and new intractability
results. Our tractability results show that $Z_{\mathrm{Ising}}(-; \beta)$ has
an FPTAS when $\lvert \beta - 1 \rvert / \lvert \beta + 1 \rvert < \tan(\pi /
(4 \Delta - 4))$. The core of the proof is showing that there are no inputs~$G$
that make the partition function $0$ when $\beta$ is in this range. Our result
significantly extends the known zero-free region of the Ising model (and hence
the known approximation results). Our intractability results show that it is
$\mathrm{\#P}$-hard to multiplicatively approximate the norm and to additively
approximate the argument of $Z_{\mathrm{Ising}}(-; \beta)$ when $\beta \in
\mathbb{C}$ is an algebraic number such that $\beta \not \in \mathbb{R} \cup
\{i,-i\}$ and $\lvert \beta - 1\rvert / \lvert \beta + 1 \rvert > 1 /
\sqrt{\Delta - 1}$. These are the first results to show intractability of
approximating $Z_{\mathrm{Ising}}(-, \beta)$ on bounded degree graphs with
complex $\beta$. Moreover, we demonstrate situations in which zeros of the
partition function imply hardness of approximation in the Ising model.
