I've recently been learning some complex geometry, mostly for my own edification. In the course of my studies I came across the following statement:
If $X$ is a Riemann surface then a choice of complex structure is equivalent to choice of conformal structure.
That is, complex structures and conformal structures on a Riemann surface are in bijection.
I haven't seen an analog of this statement for higher dimensional manifolds. Given that the intuition that one develops in 1-complex variable falls apart alarmingly fast in several complex variables, I suspect that complex and conformal structures are not equivalent in higher dimensions.
My question is as follows. Are complex structures and conformal structures inequivalent in dimension >1? If so how can I understand this (why aren't they)? Furthermore, does the correspondence fall apart in any controlled way. For instance does conformal structure imply a complex structure but that the converse no longer holds? Or are conformal and complex structures simply incomparable in higher dimensions?