Let $X$ be a complex manifold and let $$f\_X^n : \bigoplus\_{p + q = n} H^{p, q}(X) \to H^n\_{\mathrm{DR}}(X, \, \mathbb{C})$$ be the natural map from the Dolbeault cohomology to the de Rham cohomology with complex coefficients. 



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> **Question**. Is there an example of a compact complex manifold $X$ which is not Kähler such that $f\_X^n$ is an isomorphism for all $n$? 
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Examples of non-Kähler compact complex manifolds which satisfy the Dolbeault isomorphism
A compact complex manifold $X$ satisfies the Hodge decomposition $$H^k\_{\mathrm{DR}}(X, \, \mathbb C) = \bigoplus\_{p+q=k}H^{p, q}(X)$$ (possibly without Hodge symmetry) if its Frölicher spectral sequence 
$$E\_1^{p,q} = H^{p,q}(X) \Rightarrow H^{p+q}(X)$$
degenerates at $E\_1$. This happens for instance if $X$ satisfies the $\partial \bar{\partial}$-lemma. 


All Moishezon manifolds satisfy it, because the $\partial \bar{\partial}$-lemma is true for projective manifolds, so it is true for any bimeromorphic modification of them. Furthermore, a Moishezon manifold admits a Kähler metric if and only if it is projective.


Thus, in order to obtain the example you are looking for, it suffices to take a non-projective Moishezon manifold (which only exists in dimension $\geq 3$). 


For more details and references, you can look at


 Dan Popovici, [**Deformation openness and closedness of various classes of compact complex manifolds; examples**](http://www.ams.org/mathscinet-getitem?mr=3235516). *Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)* **13** (2014), no. 2, 255--305.


