  In this work we study topological properties of the one-dimensional Creutz
ladder model with different non-Hermitian asymmetric hoppings and on-site
imaginary potentials, and obtain phase diagrams regarding the presence and
absence of an energy gap and in-gap edge modes. The non-Hermitian skin effect
(NHSE), which is known to break the bulk-boundary correspondence (BBC), emerges
in the system only when the non-Hermiticity induces certain unbalanced
non-reciprocity along the ladder. The topological properties of the model are
found to be more sophisticated than that of its Hermitian counterpart, whether
with or without the NHSE. In one scenario without the NHSE, the topological
winding is found to exist in a two-dimensional plane embedded in a
four-dimensional space of the complex Hamiltonian vector. The NHSE itself also
possesses some unusual behaviors in this system, including a high spectral
winding without the presence of long-range hoppings, and a competition between
two types of the NHSE, with the same and opposite inverse localization lengths
for the two bands, respectively. Furthermore, it is found that the NHSE in this
model does not always break the conventional BBC, which is also associated with
whether the band gap closes at exceptional points under the periodic boundary
condition.
