  The Aharonov-Bohm (AB) cage enables localized confinement with nondiffractive
propagation for arbitrary excitation. In this study, we introduce an
anti-parity-time (anti-$\mathcal{PT}$) symmetric imaginary coupling in a
generalized Creutz ladder to construct a non-Hermitian AB cage with tunable
flat-band energy. We investigate compact localized states and complete
localization dynamics, and show that non-Hermiticity affects the localization
probability distributions and increases the oscillation period of the AB cage
dynamics. Non-Hermitian engineering of the decoupled core of the AB cage is the
essential point in our proposal. Our approach is widely applicable to a more
general situation and can facilitate the manipulation of localization in
physics.
