  We construct a basis of conformal primary wavefunctions (CPWs) for $p$-form
fields in any dimension, calculating their scalar products and exhibiting the
change of basis between conventional plane wave and CPW mode expansions. We
also perform the analysis of the associated shadow transforms. For each family
of $p$-form CPWs, we observe the existence of pure gauge wavefunctions of
conformal dimension $\Delta=p$, while shadow $p$-forms of this weight are only
pure gauge in the critical spacetime dimension value $D=2p+2$. We then provide
a systematic technique to obtain the large-$r$ asymptotic limit near $\mathscr
I$ based on the method of regions, which naturally takes into account the
presence of both ordinary and contact terms on the celestial sphere. In $D=4$,
this allows us to reformulate in a conformal primary language the links between
scalars and dual two-forms.
