  Recently, many novel and exotic phases have been proposed by considering the
role of topology in non-Hermitian systems, and their emergent properties are of
wide current interest. In this work we propose the non-Hermitian generalization
of semi-Dirac semimetals, which feature a linear dispersion along one momentum
direction and a quadratic one along the other. We study the topological phase
transitions in such two-dimensional semi-Dirac semimetals in the presence of a
particle gain-and-loss term. We show that such a non-Hermitian term creates
exceptional points originating out of each semi-Dirac point. We map out the
topological phase diagram of our model, using winding number and vorticity as
topological invariants of the system. By means of numerical and analytical
calculations, we examine the nature of edge states for different types of
semi-Dirac models and establish bulk-boundary correspondence and absence of the
non-Hermitian skin effect, in one class. On the other hand, for other classes
of semi-Dirac models with asymmetric hopping, we restore the non-Hermitian skin
effect, an anomalous feature usually present in non-Hermitian topological
systems.
