The proof that flat maps are open in the complex-analytic category can be found in

C. Bănică, O. Stănăşilă: *[Algebraic methods in the global theory of complex spaces][1]*, Theorem 2.12 p. 180.

See also my answer to [MO question 41158][2].


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=463470
  [2]: https://mathoverflow.net/questions/41158/are-flat-morphisms-of-analytic-spaces-open/41163#41163