The proof for opennes of flat maps in the complex-analytic category can be found in

Banica-Stanasila, *Algebraic methods in the global theory of Complex Spaces*, Theorem 2.12 p. 180.

See my answer to [MO question 41158][1].


  [1]: https://mathoverflow.net/questions/41158/are-flat-morphisms-of-analytic-spaces-open/41163#41163