Weyl semimetal is an archetypical gapless topological phase of matter. Its bulk dispersion contains pairs of band degeneracy points, or Weyl points, that act as magnetic monopoles in momentum space and lead to Fermi arc surface states. It also realizes chiral anomaly first discovered in quantum field theory: parallel electric and magnetic fields generate a finite chiral current. Here, we introduce a minimal model for non-Hermitian Weyl semimetal, dubbed point-gap Weyl semimetal, where a pair of Weyl points are located on the imaginary axis of the complex energy plane. We show the generalization triggers a few fundamental changes to the topological characterization and response of Weyl semimetals. The non-Hermitian system is characterized by a new point-gap invariant $W_3$, giving rise to complex Fermi arc surface states that cover the point gap area $W_3$ times. The splitting of Weyl points on the complex energy plane also leads to anisotropic skin effect as well as a novel type of boundary-skin modes in wire geometry. A unique feature of point-gap Weyl semimetal is a time-dependent electric current flowing along the direction of the magnetic field in the absence of electric field, due to the chiral imbalance created by the different lifetime of the Weyl fermions. We discuss the experimental signatures in wave-packet dynamics and possible realizations of point-gap Weyl semimetal in synthetic platforms.
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