This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme that a Calabi-Yau $n$-manifold (compact Kähler with $h^{k,0} = 0$ for $0 < k < n$, trivial canonical bundle) is homeomorphic to a non-Calabi-Yau complex manifold?
In other words, could there be a topological manifold that admits one complex structure for which it is Calabi-Yau, and another for which it isn't?