  A quantum system governed by a non-Hermitian Hamiltonian may exhibit zero
temperature phase transitions that are driven by interactions, just as its
Hermitian counterpart, raising the fundamental question how non-Hermiticity
affects quantum criticality. In this context we consider a non-Hermitian system
consisting of an $\rm XY$ model with a complex-valued four-state clock
interaction that may or may not have parity-time-reversal ($\mathcal{PT}$)
symmetry. When the $\mathcal{PT}$ symmetry is broken, and time-evolution
becomes non-unitary, a scaling behavior similar to the
Berezinskii-Kosterlitz-Thouless phase transition ensues, but in a highly
unconventional way, as the line of fixed points is absent. From the analysis of
the $d$-dimensional RG equations, we obtain that the unconventional behavior in
the $\mathcal{PT}$ broken regime follows from the collision of two fixed points
in the $d\to 2$ limit, leading to walking behavior or pseudocriticality. For
$d=2+1$ the near critical behavior is characterized by a correlation length
exponent $\nu=3/8$, a value smaller than the mean-field one. These results are
in sharp contrast with the $\mathcal{PT}$-symmetric case where only one fixed
point arises for $2<d<4$ and in $d=1+1$ three lines of fixed points occur with
a continuously varying critical exponent $\nu$.
