  This paper is devoted to studying the conformal Willmore functional for
surfaces in $4$-dimensional conformal manifolds. We calculate the first and
second variation. The Euler-Lagrange equation of this functional is stated in a
conformal invariant form. Based on the second variation formula we obtained, we
prove that the Clifford torus in $\mathbb{C}P^2$ is strongly Willmore-stable.
This provides strong support to the conjecture of Montiel and Urbano, which
states that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore
functional among all tori. In $4$-dimensional locally symmetric spaces, by
constructing several holomorphic differentials, we prove that among all minimal
$2$-spheres only those super-minimal ones can be Willmore.
