  A C-symplectic structure is a complex-valued 2-form which is holomorphically
symplectic for an appropriate complex structure. We prove an analogue of
Moser's isotopy theorem for families of C-symplectic structures and list
several applications of this result. We prove that the degenerate twistorial
deformation associated to a holomorphic Lagrangian fibration is locally trivial
over the base of this fibration. This is used to extend several theorems about
Lagrangian fibrations, known for projective hyperk\"ahler manifolds, to the
non-projective case. We also exhibit new examples of non-compact complex
manifolds with infinitely many pairwise non-birational algebraic
compactifications.
