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: