  In this study, we explore the interplay between $\mathcal{PT}$-symmetry and
quantum chaos in a non-Hermitian dynamical system. We consider an extension of
the standard diagnostics of quantum chaos, namely the complex level spacing
ratio and out-of-time-ordered correlators (OTOCs), to study the
$\mathcal{PT}$-symmetric quantum kicked rotor model. The kicked rotor has long
been regarded as a paradigmatic dynamic system to study classical and quantum
chaos. By introducing non-Hermiticity in the quantum kicked rotor, we uncover
new phases and transitions that are absent in the Hermitian system. From the
study of the complex level spacing ratio, we locate three regimes -- one which
is integrable and $\mathcal{PT}$-symmetry, another which is chaotic with
$\mathcal{PT}$-symmetry and a third which is chaotic but with broken
$\mathcal{PT}$-symmetry. We find that the complex level spacing ratio can
distinguish between all three phases. Since calculations of the OTOC can be
related to those of the classical Lyapunov exponent in the semi-classical
limit, we investigate its nature in these regimes and at the phase boundaries.
In the phases with $\mathcal{PT}$-symmetry, the OTOC exhibits behaviour akin to
what is observed in the Hermitian system in both the integrable and chaotic
regimes. Moreover, in the $\mathcal{PT}$-symmetry broken phase, the OTOC
demonstrates additional exponential growth stemming from the complex nature of
the eigenvalue spectrum at later times. We derive the analytical form of the
late-time behaviour of the OTOC. By defining a normalized OTOC to mitigate the
effects caused by $\mathcal{PT}$-symmetry breaking, we show that the OTOC
exhibits singular behaviour at the transition from the $\mathcal{PT}$-symmetric
chaotic phase to the $\mathcal{PT}$-symmetry broken, chaotic phase.
