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.

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.

EDIT. I forgot to mention this:

B. Branner, K. Dias, Classification of complex polynomial vector fields in one complex variable, Journal Journal of Difference Equations and Applications Volume 16, 2010 - Issue 5-6: