  This article aims to study the existence of stable Bloch oscillations and
Landau-Zener tunneling in a non-Hermitian system when exposed to external
fields. We investigate a non-Hermitian $\cal{PT}$-symmetric diamond chain
network and its transport dynamics in two different situations, namely in a
flat band case and a non-flat band case. The considered system does not support
unbroken-$\cal{PT}$ phase or completely real eigenspectra in any of the
parametric regions in both the flat and non-flat band cases. In the flat band
case, up to a critical value of the gain-loss parameter, the bands are found to
be gapless or inseparable, and for other values the bands are isolated.
Considering the non-flat band case, all the bands are found to be complex
dispersive and are also isolated. In the case of completely broken $\cal{PT}$
phase, we look upon the possibility to have stable dynamics or Bloch
oscillations upon the application of external fields like synthetic electric
field. In particular, when the complex bands are isolated, we point out that
the Landau-Zener tunneling induced by the synthetic electric field can enable
Bloch oscillations. The amplitude of these Bloch oscillations is large and
persists for a long propagation distance which reveals that super Bloch
oscillations can be observed in the broken $\cal{PT}$ phase of the system. We
also report the amplified Bloch oscillations which pave the way towards
controlling transport phenomena in non-Hermitian systems.
