I will recall some of the difficulties in realizing quantization as a functor. In the framework of geometric quantization, one of such difficulties is associated with the dependence of quantization on the choice of a complex structure on the symplectic manifold. I will describe some natural infinite dimensional families of Kaehler structures degenerating to Lagrangian fibrations on cotangent bundles of compact Lie groups and to the singular $T^n$ fibration on toric manifolds. Coming from the other side these families of Kaehler structures can be viewed as complex time evolutions of the Lagrangian fibrations. This point of view, together with an appropriate uniqueness theorem, allows us to shed new light on the coherent state transform for compact Lie groups and on its relation with geometric quantization. This talk is based on joint work in progress with Will Kirwin and João P. Nunes.