We propose an adaptive finite element method for the elliptic interface problem with piecewise constant coefficients, based on a robust variant of the original NXFEM of Hansbo and Hansbo [4], whitch allows for discontinuities, internal to the elements, in the approximation across the interface. This robust variant, introducing coefficients pondaration, provides constants in the error estimates independent of the geometry of the cut cells and also the coefficients. We extend the method to quadrilateral meshes and derive error estimators and perform several numerical tests.