We study the complex dynamics in analytic area-preserving maps in a neighbourhood of a resonant elliptic fixed point. We assume that the resonance is weak, i.e., the linear part is a rotation of an angle 2\pi where n\geq5. Normal form theory suggests that there is a flower with n petals which consists of points bi-asymptotic to the fixed point. We show that the flower splits: there are parabolic stable and unstable complex manifolds and they do not intersect. We measure the splitting of the manifolds and relate it to the Stokes phenomenon. This study is relevant for describing the splitting of the separatrices in the elliptic islands which appear in the generic unfolding of a resonant elliptic fixed point.