  In this paper, we investigate the influence of quasiperiodic perturbations on
one-dimensional non-Hermitian diamond lattices with an artificial magnetic flux
$\theta$ that possess flat bands. Our study shows that the symmetry of these
perturbations and the magnetic flux $\theta$ play a pivotal role in shaping the
localization properties of the system. When $\theta=0$, the non-Hermitian
lattice exhibits a single flat band in the crystalline case, and symmetric as
well as antisymmetric perturbations can induce accurate mobility edges. In
contrast, when $\theta=\pi$, the clean diamond lattice manifests three
dispersionless bands referred to as an "all-band-flat" (ABF) structure,
irrespective of the non-Hermitian parameter. The ABF structure restricts the
transition from delocalized to localized states, as all states remain localized
for any finite symmetric perturbation. Our numerical calculations further
unveil that the ABF system subjected to antisymmetric perturbations exhibits
multifractal-to-localized edges. Multifractal states are predominantly
concentrated in the internal region of the spectrum. Additionally, we explore
the case where $\theta$ lies within the range of $(0, \pi)$, revealing a
diverse array of complex localization features.
