Deligne complex $A(p)$ is the complex

$$A((2\pi i)^p) \rightarrow \mathcal O \rightarrow \Omega^1 \rightarrow {}...{} \rightarrow \Omega^{p-1}.$$

You can work with similar bigraded complexes, taking into account not only holomorphic, but also antiholomorphic forms. Let us define $A(p,q)$ as the complex of sheaves

$$A((2\pi i)^{p+q}) \rightarrow \mathcal O \oplus \overline{\mathcal O} \rightarrow \Omega^1\oplus \overline{\Omega^1} \rightarrow {}...{} \rightarrow \Omega^{p-1} \oplus \overline{\Omega^{p-1}} \rightarrow \overline{\Omega^p} \rightarrow ... \rightarrow \overline{\Omega^{q-1}}.$$

(Here it's supposed that $q>p$). In this situation, we may take $A$ to be a subring of $\Bbb C$, not necessarily of $\Bbb R$. Suppose that $A = \Bbb C$.
Then we can write a resolution of this complex of sheaves by differential forms, and it's possible to see that $A(p,q)$ is quasiisomorphic to the complex

$$ ... \stackrel{d}{\rightarrow} \mathcal{A}^{p-2, q-1} \oplus \mathcal{A}^{p-1, q-2} \stackrel{d}{\rightarrow}
\mathcal{A}^{p-1, q-1} \stackrel{\partial\bar\partial}{\rightarrow}
\mathcal{A}^{p, q} \stackrel{d}{\rightarrow} \mathcal{A}^{p, q-1} \oplus \mathcal{A}^{p-1, q} \stackrel{d}{\rightarrow}{} ...$$

where the term $\mathcal{A}^{p-1, q-1}$ has the grading $p+q-2$.
(The $\partial\bar\partial$ operator arises because of local $\partial\bar\partial$-lemma. You can find the proof that these complexes are quasiisomorphic in M. Schweitzer's text https://arxiv.org/abs/0709.3528).

In particular, $\Bbb H^{p+q-2}(X,{~}A(p,q))$ is known as Aeppli cohomology
$H_{Ae}^{p-1,q-1}$, and $\Bbb H^{p+q-1}(X,{~}A(p,q))$ is Bott-Chern cohomology $H_{BC}^{p,q}$. If your $X$ satisfies global $\partial\bar\partial$-lemma (for example, if it is compact Kaehler), then these are
just isomorphic to the corresponding Dolbeaut cohomology; on a general complex manifold, they are different.

The Deligne complex with real coefficients $\Bbb R(p)$ is isomorphic to the fixed points of complex conjugation on $\Bbb R(p,p)$, so your can take real forms in the Aeppli-Bott-Chern resolution and obtain the resolution for the real Deligne cohomology.

Now, from the looks of the resolution and a bit of harmonic theory one can conclude that there's a non-degenerate pairing between
$\Bbb H^{k}(X,{~}\Bbb C(p,q))$ and $\Bbb H^{2n+1-k}(X,{~}\Bbb C(n+1-p,n+1-q))$ (at least when $X$ is compact, otherwise you need to take cohomology with compact support on one side). This is complex version of the duality on Deligne cohomology, and when $k=p+q$ it specializes to the well-known duality between Bott-Chern and Aeppli cohomology (and to Poincare duality when $X$ satisfies $\partial\bar\partial$-lemma).

I don't know what happens if you work with Deligne-Beilinson complex instead of Deligne complex (with logarithmic forms instead of regular), but I'd suppose that everything works the same and gives you the duality between Bott-Chern cohomology and Aeppli cohomology with compact support on an open part of your variety as a particular case.