Here is a philosophical idea. exploit the following asymmetry in our state of knowledge about closed orientable manifolds: whereas almost complex is equivalent to almost symplectic: symplectic entails a further homological condition while being complex entails no further known homological condition.
This potential unknown further condition to be a closed complex manifold must reduce to no condition in complex dimension one as does the symplectic condition reduce to no condition in complex dimension one. Looking at known examples one can ask whether for a closed complex manifold above complex dimension one must the sum of the betti numbers necessarily be at least three. Note complex projective two space realizes three and above complex dimension two one has the circle cross odd spheres with total betti number four. So the guess is close to being sharp if true, and if true proves only the two sphere among manifolds with the betti numbers of the even sphere can be a closed complex manifold. Note: For manifolds of even complex dimension the last statement has been known since the work of rene thom on cobordism, the signature and the euler characteristic showing there is not even an almost complex structure.