  We study spectral approximations of Schr\"odinger operators $T=-\Delta+Q$
with complex potentials on $\Omega=\mathbb{R}^d$, or exterior domains
$\Omega\subset \mathbb{R}^d$, by domain truncation. Our weak assumptions cover
wide classes of potentials $Q$ for which $T$ has discrete spectrum, of
approximating domains $\Omega_n$, and of boundary conditions on $\partial
\Omega_n$ such as mixed Dirichlet/Robin type. In particular, ${\rm Re} \, Q$
need not be bounded from below and $Q$ may be singular. We prove generalized
norm resolvent convergence and spectral exactness, i.e. approximation of all
eigenvalues of $T$ by those of the truncated operators $T_n$ without spectral
pollution. Moreover, we estimate the eigenvalue convergence rate and prove
convergence of pseudospectra. Numerical computations for several examples, such
as complex harmonic and cubic oscillators for $d=1,2,3$, illustrate our
results.
