  It has been recently shown that complex two-dimensional (2D) potentials
$V_\varepsilon(x,y)=V(y+\mathrm{i}\varepsilon\eta(x))$ can be used to emulate
non-Hermitian matrix gauge fields in optical waveguides. Here $x$ and $y$ are
the transverse coordinates, $V(y)$ and $\eta(x)$ are real functions,
$\varepsilon>0$ is a small parameter, and $\mathrm{i}$ is the imaginary unit.
The real potential $V(y)$ is required to have at least two discrete eigenvalues
in the corresponding 1D Schr\"odinger operator. When both transverse directions
are taken into account, these eigenvalues become thresholds embedded in the
continuous spectrum of the 2D operator. Small nonzero $\varepsilon$ corresponds
to a non-Hermitian perturbation which can result in a bifurcation of each
threshold into an eigenvalue. Accurate analysis of these eigenvalues is
important for understanding the behavior and stability of optical waves
propagating in the artificial non-Hermitian gauge potential. Bifurcations of
complex eigenvalues out of the continuum is the main object of the present
study. We obtain simple asymptotic expansions in $\varepsilon$ that describe
the behavior of bifurcating eigenvalues. The lowest threshold can bifurcate
into a single eigenvalue, while every other threshold can bifurcate into a pair
of complex eigenvalues. These bifurcations can be controlled by the Fourier
transform of function $\eta(x)$ evaluated at certain isolated points of the
reciprocal space. When the bifurcation does not occur, the continuous spectrum
of 2D operator contains a quasi-bound-state which is characterized by a
strongly localized central peak coupled to small-amplitude but nondecaying
tails. The analysis is applied to the case examples of parabolic and
double-well potentials $V(y)$. In the latter case, the bifurcation of complex
eigenvalues can be dampened if the two wells are widely separated.
