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In this paper, we investigate the fluctuations of a unit eigenvectorassociated to an outlier in the spectrum of a spiked $N\times N$ complexDeformed Wigner matrix $M_N$: $M_N =W_N/\sqrt{N} + A_N$ where $W_N$ is an $N\times N$ Hermitian Wigner matrix whose entries have a law $\mu$ satisfying aPoincar\'e inequality and the matrix $A_N$ is a block diagonal matrix, with aneigenvalue $\theta$ of multiplicity one, generating an outlier in the spectrumof $M_N$. We prove that the fluctuations of the norm of the projection of aunit eigenvector corresponding to the outlier of $M_N$ onto a unit eigenvectorcorresponding to $\theta$ are not universal.

Non universality of fluctuations of outlier eigenvectors for block diagonal deformations of Wigner matrices