•The Abel differential equation has been applied to modelize problems from ecology, control theory, electrical circuits, cosmology,...•The interest for understanding the dynamics of this equation is proved by the hundreds of papers dedicated to it in MathSciNet.•In this paper we characterize the phase portraits of the complex Abel polynomial differential equation z′=(z−a)(z−b)(z−c).•The real version of this equation depends on six parameters and consequently the classification of their phase portraits needs some work. In this paper we characterize the phase portraits of the complex Abel polynomial differential equationsz˙=(z−a)(z−b)(z−c),with z∈C, a,b,c∈C. We give the complete description of their topological phase portraits in the Poincaré disc, i.e. in the compactification of R2 adding the circle S1 of the infinity.