As in the case of irreducible holomorphic symplectic manifolds, the period domain Compl of compact complex tori of even dimension 2n contains twistor lines. These are special 2-spheres parametrizing complex tori whose complex structures arise from a given quaternionic structure. In analogy with the case of irreducible holomorphic symplectic manifolds, we show that the periods of any two complex tori can be joined by a generic chain of twistor lines. We also prove a criterion of twistor path connectivity of loci in Compl where a fixed second cohomology class stays of Hodge type (1,1). Furthermore, we show that twistor lines are holomorphic submanifolds of Compl, of degree 2n in the Plücker embedding of Compl.