  Phase transitions in non-Hermitian systems are at the focus of cutting edge
theoretical and experimental research. On the one hand, parity-time- ($\cal
PT$-) and anti-$\cal PT$-symmetric physics have gained ever-growing interest,
due to the existence of non-Hermitian spectral singularities called exceptional
points (EPs). On the other, topological and localization transitions in
non-Hermitian systems reveal new phenomena, e.g., the non-Hermitian skin effect
and the absence of conventional bulk-boundary correspondence. The great
majority of previous studies exclusively focus on non-Hermitian Hamiltonians,
whose realization requires an {\it a priori} fine-tuned extended lattices to
exhibit topological and localization transition phenomena.In this work, we show
how the non-Hermitian localization phenomena can naturally emerge in the
synthetic field moments space of zero-dimensional bosonic systems, e.g., in
anti-$\cal PT$ and $\cal PT$-symmetric quantum dimers. This offers an
opportunity to simulate localization transitions in low-dimensional systems,
without the need to construct complex arrays of, e.g., coupled cavities or
waveguides. Indeed, the field moment equations of motion can describe an
equivalent (quasi-)particle moving in a one-dimensional (1D) synthetic lattice.
This synthetic field moments space can exhibit a nontrivial localization
phenomena, such as non-Hermitian skin effect, induced by the presence of
highly-degenerate EPs. We demonstrate our findings on the example of an
anti-$\cal PT$-symmetric two-mode system, whose higher-order field moments
eigenspace is emulated by a synthetic 1D non-Hermitian Hamiltonian having a
Sylvester matrix shape. Our results can be directly verified in
state-of-the-art optical setups, such as superconducting circuits and toroidal
resonators, by measuring photon moments or correlation functions.
