A connected compact complex Lie group is a torus, hence the question is apparently about a principal torus bundles. Of course, product of a torus and a compact Kahler manifold is Kahler, giving a trivial answer to your question. For non-trivial examples, one may take a finite quotient of a trivial torus fibration by a finite group acting freely and compatible with the fibration structure.

A connected compact complex Lie group is a torus, hence the question is apparently about principal torus bundles. Of course, product of a torus and a compact Kahler manifold is Kahler, giving a trivial answer to your question. For non-trivial examples, one may take a finite quotient of a trivial torus fibration by a finite group acting freely and compatible with the fibration structure.