Complexity equals anything for multi-horizon black holes

We investigate the ``complexity equals anything" proposal with codimension-one and codimension-zero gravitational observables for multi-horizon black holes, using the Bardeen-AdS class black hole as an example. In particular, we compare the results with the ``complexity equals volume" (CV) proposal and find that the ``complexity equals anything" ($\mathcal{C}_{\text{gen}}$) enables the probing of a more complete black hole interior, that is, all spacetime regions where the blackening factor $f(r)<0$. This is the advantage brought by the flexibility of this holographic complexity conjecture. In addition, we compute the complexity derived from various geometric quantities and show that these constructions can effectively differentiate the distinct interior regions of the black hole.