In holomorphic symplectic geometry we often study the situation when a symplectic manifold admits a proper map to some other variety of positive dimension and with positive-dimensional fibres. A well known result of Matsushita tells that the fibres of such a fibration are always Lagrangian, and the smooth fibres are abelian varieties. This leads to the notion of a holomorphic Lagrangian fibration - one of the central subjects of symplectic geometry. Given a holomorphic Lagrangian fibration one can deform the symplectic form and the complex structure of the original symplectic manifold, producing the so-called degenerate twistor deformation. In the talk I will try to explain my recent joint work with Misha Verbitsky, where we study the properties of such deformations. We prove that for a holomorphic Lagrangian fibration on a compact hyperkähler manifold the degenerate twistor deformations are always Kähler, confirming a conjecture of Markman.