  Higher-order topology in non-Hermitian (NH) systems has recently become one
of the most promising and rapidly developing fields in condensed matter
physics. Many distinct phases that were not present in the Hermitian
equivalents are shown in these systems. In this work, we examine how
higher-order Weyl semimetals are impacted by NH perturbation. We identify a new
type of topological semimetal, i.e., non-Hermitian higher-order Weyl semimetal
(NHHOWS) with surface diabolic points. We demonstrate that in such an NHHOWS,
new exceptional points inside the bulk can be created and annihilated,
therefore allowing us to manipulate their number. At the boundary, these
exceptional points are connected through unique surface states with diabolic
points and hinge states. For specific system parameters, the surface of NHHOWS
behaves as a Dirac phase with linear dispersion or a Luttinger phase with a
quadratic dispersion, thus paving a way for Dirac-Luttinger switching. Finally,
we employ the biorthogonal technique to reinstate the standard bulk boundary
correspondence for NH systems and compute the topological invariants. The
obtained quantized biorthogonal Chern number and quadruple moment topologically
protect the unique surface and hinge states, respectively.
