Does anyone have an example or know any references for a complex manifold $M$ with two different holomorphic structures that give rise to different Hodge numbers?
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$\begingroup$ pleasantpheasant  how do you define the Hodge numbers in the nonK\"ahler case? $\endgroup$– algoriCommented Jul 13 at 19:52

3$\begingroup$ algori  The dimension of the Dolbeault cohomology groups on M $\endgroup$– pleasantpheasantCommented Jul 13 at 19:55

4$\begingroup$ In a complex manifold the holomorphic structure is fixed. You probably mean "a smooth manifold $M$ with two different complex structures"? $\endgroup$– YCorCommented Jul 14 at 4:55
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1 Answer
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A wellstudied example is given by the Iwasawa manifold, see 1.4.3 in Angella's thesis here https://arxiv.org/abs/1302.0524.