Monodromy problems are a classical topic in the theory of ODEs on complex domains. We will describe an approach to the study of certain rank 2 ODEs on Riemann surfaces via branched complex projective structures on them, i.e. geometric structures which are locally modelled on the geometry of the Riemann sphere, possibly with integral conical singularities. These structures admit a quite rich deformation theory, and wewill focus on the geometric surgeries needed to describe structures corresponding to quasi-Fuchsian holonomy representation.