Yes, they have been studied. Look into the recent paper Langley, J. K. Trajectories escaping to infinity in finite time. Proc. Amer. Math. Soc. 145 (2017), no. 5, 2107–2117, and the reference list in this paper.
They were also studied by physicists:
Bender, Carl M.; Hook, Daniel W. Complex classical motion in potentials with poles and turning points. Stud. Appl. Math. 133 (2014), no. 3, 318–336.