Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor.
Does the same thing hold for compact complex manifolds? i.e.,
(1) Does any compact complex manifold have a dense Stein open subset?
(2) If (1) is true, can one take a dense Stein open subset as the complement of a divisor?