Is the Bott-Chern/Aeppli cohomology determined by the de Rham and Dolbeault cohomologies? Is there an example of two different complex manifolds that have the same de Rham cohomology and Dolbeault cohomology but different Bott-Chern/Aeppli cohomology?
 A: As for compact complex surfaces, the Bott-Chern/Aeppli cohomologies are determined by de Rham/Dolbeault cohomologies:this is contained in Lemma 2.3 in:
Teleman, A.: The pseudo-effective cone of a non-Kählerian surface and applications. Math. Ann. 335(4), 965–989 (2006), arXiv:0704.2948.
A: I learnt all of the following from section $3.2$ of Angella's Cohomological Aspects in Complex Non-Kähler Geometry.

Let $R$ be a commutative ring with identity. The three-dimensional Heisenberg group over $R$ is
$$\mathbb{H}(3, R) = \left\{\begin{bmatrix} 1 & z^1 & z^3\\ 0 & 1 & z^2\\ 0 & 0 & 1\end{bmatrix} : z^1, z^2, z^3 \in R\right\}.$$
The Iwasawa manifold $\mathbb{I}_3$ is the quotient of $\mathbb{H}(3, \mathbb{C})$ by the discrete subgroup $\mathbb{H}(3, \mathbb{Z}[i])$ acting on the left, i.e. $\mathbb{I}_3 := \mathbb{H}(3, \mathbb{Z}[i])\setminus\mathbb{H}(3, \mathbb{C})$. It is a compact complex three-dimensional manifold which is holomorphically parallelisable and not of Kähler type. 
The small deformations of the Iwasawa manifold were classified by Nakamura in Complex Parallelisable Manifolds and their Small Deformations. There are three classes, corresponding to the possible Hodge numbers of the deformations. Nakamura explicitly determined which small deformations gave rise to which class. Furthermore, the conjugate Hodge numbers are constant within a given class and they differ between classes. 
Angella showed that the deformations of class (ii) further decompose into two subclasses: (ii.a) and (ii.b). The Bott-Chern numbers and Aeppli numbers are constant within a given subclass, but they differ between subclasses. In particular, if $X_a$ and $X_b$ are small deformations of $\mathbb{I}_3$ in subclasses (ii.a) and (ii.b) repectively, then 
$$h^{2,2}_{\text{BC}}(X_a) = 7,\qquad h^{2,2}_{\text{BC}}(X_b) = 6,\qquad h^{p,q}_{\text{BC}}(X_a) = h^{p,q}_{\text{BC}}(X_b)\ \text{for}\ (p,q) \neq (2, 2).$$
The Hodge star operator gives rise to an isomorphism between $(p, q)$ Bott-Chern cohomology and $(n-q, n-p)$ Aeppli cohomology, so $h^{p,q}_{\text{BC}} = h^{n-q,n-p}_{\text{A}}$. Therefore, relationship between the Aeppli numbers of $X_a$ and $X_b$ is
$$h^{1,1}_{\text{A}}(X_a) = 7,\qquad h^{1,1}_{\text{A}}(X_b) = 6,\qquad h^{p,q}_{\text{A}}(X_a) = h^{p,q}_{\text{A}}(X_b)\ \text{for}\ (p,q) \neq (1, 1).$$
So, $X_a$ and $X_b$ have different Bott-Chern numbers and different Aeppli numbers, but as $X_a$ and $X_b$ are both deformations of class (ii), they have the same Hodge numbers (and conjugate Hodge numbers). Finally, by Ehresmann's Theorem, $X_a$ and $X_b$ are diffeomorphic so they have the same Betti numbers.
