It is well known that in dimension $3$ and higher there exist complex structures on diffeomerphic manifolds with totally different Chern classes (and Chern numbers).


For the case of complex manifolds you can check 

https://mathoverflow.net/questions/26586/can-one-bound-the-todd-class-of-a-3-dimensional-variety-polynomially-in-c-3/26598#26598

For the case of complex projective manifolds the reference given in the same answer: 

http://arxiv.org/PS_cache/arxiv/pdf/0903/0903.1587v1.pdf