Let f be a dominant polynomial transformation of the complex affine plane. The dynamical degree lambda_1 of f is defined as the limit of the n-th root of the degree of the n-th iterate of f. In 2007, Favre and Jonsson showed that the dynamical degree of any polynomial endomorphism of the affine plane is a quadratic integer. For any affine surface S0, there is a definition of the dynamical degree that generalizes the one on the affine plane. We show that the result still holds for any complex affine surface: the dynamical degree of an endomorphism of any complex affine surface is a quadratic integer. The proof uses the space of valuations centered at infinity V. The endomorphism f defines a transformation of V and studying the dynamics of f on V gives information about the dynamics of f on S0. The main result is that under certain hypothesis, f admits an attracting fixed point in V that we call an eigenvaluation. This implies that one can find a good compactification S of S0 such that f admits an attracting fixed point p at infinity and f has a normal form at p; the result on the dynamical degree follows from the normal form.