  We outline some general features of possible extensions of the Standard Model
that include anomalous U(1) gauge symmetries, a certain number of axions and
their mixings with the CP-odd Higgs sector. As previously shown, after the
mixing one of the axions becomes a physical pseudoscalar (the axi-Higgs) that
can take the role of a modified QCD axion. It can be driven to be very light by
the same non-perturbative effects that are held responsible for the solution of
the strong CP-problem. At the same time the axi-Higgs has a sizeable gauge
interaction, which is not allowed to the Peccei-Quinn axion, possibly
explaining the PVLAS results. We point out that the Wess-Zumino term, typical
of these models, can be both interpreted as an anomaly inflow from higher
dimensional theories (second window) but also as a result of partial decoupling
of an extra Higgs sector (and of a fermion) that leaves behind an effective
anomalous abelian theory (first window) in a broken St\"{u}ckelberg phase. The
possibility that the axi-Higgs can be heavy, of the order of the Higgs mass or
larger, however, can't be excluded. The potentialities for the discovery of
this particle and of anomaly effects in the neutral current sector at the LHC
are briefly discussed in the context of a superstring inspired model (second
window), but with results that remain valid also if any of the two
possibilities is realized in Nature.
