In this talk, I will present a work in progress with Matthieu Astorg and Lorena Lopez-Hernanz. We are interested in studying holomorphic endomorphisms of $C2$ which are tangent to the identity at the origin, and our goal is to understand how the dynamics changes when we perturb such maps. In particular, we generalize a result obtained by Bianchi and show a statement à la Lavaurs when the unperturbed map admits a basin parabolic centered in a characteristic direction, but it does not fix a complex line. I will recall the motivation and results in the one-dimensional case before moving to dimension 2.