First, this case is totally uninteresting regarding Hilbert XVI. Indeed, there are no limit cycles in such systems. The $\alpha / \omega$-limit of a trajectory is either a point or a non-isolated cycle (center case).
A singularity at $a\in \mathbb C$ (i.e. a root of $F$) can only be of three types, according to the value of $F'(a)$:
- Source/focus: $F'(a)\notin i\mathbb R$.
- Center: $F'(a)\in i\mathbb R_{\neq 0}$.
- Flower with $2k$ petals: $F'(a)=0$ with multiplicity $k$.
In addition there is a pole at infinity (if $\deg(F)>0$) with exactly $2\deg(F)$ separatrices, reaching the singularity in finite time. The bassins of attraction / center regions attached to the above singularities are delimited by the separatrices.
S. Smale began to get interested in the question in the early 80's while laying the foundations for BSS computational model (The fundamental theorem of algebra and complexity theory, 1981). He proposed a numerical root solver for polynomials by following the flow of $\frac{F}{F'}$. This started some works on the topic, for instance by Schub, Tischler, William (The Newtonian graph of a complex polynomial, 1988) or Benzinger (Plane autonomous systems with rational vector fields, 1991)…
In the case of these vector fields, the topological class is entirely encoded by their Newtonian graph (or the «dual» spinal graph) given by the incidence graph of the $\alpha / \omega$-limits of trajectories (in red on the picture). The main result for polynomials is that it is a tree. See e.g. Sverdlove (Inverse problems for dynamical systems,1981) and Schecter, Singer (A class of vectorfields on $\mathbb S^2$ that are topologically equivalent to polynomial vectorfields,1985) and Jongen, Jonker, Twilt (On the classification of plane graphs representing structurally stable rational Newton flows,1991).
The conformal classification has been initiated by Douady, Estrada and Sentenac (unpublished monograph, 2005) for the generic case (only focus/source singularities) and completed by Branner and Dias (Classification of complex polynomial vector fields in one complex variable, 2010). In addition to the combinatorial (topological) invariant, a complex «time-shift» (related to the integrals $\int_\gamma\frac{1}{F(z)} dz$) is associated to the separatrices, providing a complete conformal invariant.
In that latter context, the function $\int\frac{1}{F(z)} dz$ is called a Fatou coordinates. It is a rectifying chart for the vector field, and has many interesting dynamical properties.
Notice also the deep and beautiful relationship between spinal graph and Dessins d'enfants, as established by Pilgrim (Polynomial vector fields, dessins d'enfants, and circle packings,2006), related to this question.
