Using twistor techniques we shall show that there is a hypercomplex structure in the neighbourhood of the zero section of the tangent bundle TX of any complex manifold X with a real-analytic torsion-free connection compatible with the complex structure whose curvature is of type (1, 1). The zero section is totally geodesic and the Obata connection restricts to the given connection on the zero section.

We also prove an analogous result for vector bundles: any vector bundle with real-analytic connection whose curvature is of type (1, 1) over X can be extended to a hyperholomorphic bundle over a neighbourhood of the zero section of TX.