  We develop a general method to compute the Morse index of branched Willmore
spheres and show that the Morse index is equal to the index of certain matrix
whose dimension is equal to the number of ends of the dual minimal surface. As
a corollary, we find that for all immersed Willmore spheres
$\vec{\Phi}:S^2\rightarrow \mathbb{R}^3$ such that $W(\vec{\Phi})=4\pi n$, we
have $\mathrm{Ind}_{W}(\vec{\Phi})\leq n-1$
