Can a holomorphic vector field have an attractor homoclinic loop? It is  well known that a  holomorphic vector field $z'=f(z), z\in \mathbb{C}$ does not have any limit cycle.See the last paragraph of this post
Orbits space of real-analytic planar foliations
One can imagine several reason for this fact. For example:  The flow map is a holomorphic map. So if  the limit cycle is of period $T$ then $\phi_T$ admits an infinite number of fixed point hence it is identically equal to the identity map.
Now we ask the following question

Is there a holomorphic vector field $z'=f(z)$ on an open region  of $\mathbb{C}$ which admit an attractor homoclinic loop?

 A: The answer is 'no' for much the same reason that the OP indicates: the existence of a homoclinic or heteroclinic connection implies that neighboring trajectories are periodic.
First, one needs to have poles in $f$ if one wishes to have heteroclinic connections between (real) saddle singularities. Indeed if $f$ is holomorphic, then the stationnary points of the underlying real vector field are either centers, foci or multiple singularities locally conformally conjugate to $z'=\frac{z^{k+1}}{1+\mu z^k}$. None of these singularities can be part of a candidate limit (poly)cycle.
Next, consider the time-coordinate $$T(z)=\int_{z_*}^z \frac{dz}{f(z)}.$$ Real-time trajectories of the vector field correspond to level sets $\mathrm{Im}(T)=\textrm{cst}$. The presence of a (poly)cycle $\Gamma$ connecting poles of $f$ implies that $\int_\Gamma \frac{dz}{f(z)}=:\tau\in \mathbb{R}$ is the period of $\Gamma$.
Let $\gamma$ be a neighboring trajectory connecting a transverse section $\Sigma$ to itself "after making one turn". Without loss of generality one may choose $\Sigma$ as a small piece of $\mathrm{Re}(T)=\textrm{cst}$. Then there exists a small curve $\sigma\subset\Sigma$ such that the concatenation $\sigma\gamma$ is a closed loop. Cauchy's formula implies that $\int_{\sigma\gamma} \frac{dz}{f(z)}=\tau$. But only $\sigma$ contributes to the imaginary part of the integral, therefore $\sigma$ is just one point and $\gamma$ is closed.
A: There is  no any kind of, generally speaking,  "Charactristic curve" for the vector field  $z'=f(z)$  when $f$ is a holomorphic function. By charactristic curve I mean any kind of  particular curve which is really rare!. Mathematically speaking: it is  isolated in its neighborhood: there is no any thing similar to it, in its small neighborhood. For example  a limit cycle,  an attracor loop, a UINIQE hetroclinic or homoclinic  connection, etc:
>Proof: observe that $[f,if]=0$ and $f$ and $if$ are independent vector fields \QED
Remark: The above 1-line proof can be accepted via  the same argument as page 2 item 1 of my paper  https://arxiv.org/abs/math/0507516 which says: an attracting limit cycle for a vector  field $X$ is an invariant curve for  any vector field  $Y$  with $[X,Y]=0$. a common observation in investigation of limit cycles, isochronois centers, etc.
When I was asking this MO question I forgot the simple fact that $[f,if]=0$ when $f$  is a holomorphic  function. I observed it imeddiate after my first meet with the concept  "Lie bracket of vector fields" when I was a master student of the course "Differentiable manifold"(about 27 years ago). In that period I was not familliar with the concept limit cycle but it seems that "limit cycle" was in my inconscient.
Any way I lookforward to hear the answer to my commented questions to other existing answer to this question. a possible answer to my commented questions would help me to understand:
In the holomorphic setting, is it possible we have a familly of homoclinic loop $\gamma_t$ which homotopically vanish at the vertex of homoclinic loop? So this would be a contradiction to possible common belife that in the interior of a homoclinic loop we necessarilly have a singularity(of course different from the vertex).
