:It is well-known that a metallic particle can support surface plasmonic modes. For a subwavelength particle, these modes correspond to negative values of the permittivity, which are solutions of a self-adjoint eigenvalue problem. In this work, we are interested in the finite element computation of plasmonic modes in the case of a 2D particle whose boundary is smooth except for one corner. While a smooth particle has a discrete sequence of plasmonic eigenvalues, the corner leads to the presence of an essential spectrum, due to the existence of hyper-oscillating waves at the corner, the so called black-hole waves. Following our previous works, we introduce a complex scaling at the corner, and solve the complex-scaled eigenvalue problem. Numerical results show that the complex scaling deforms the essential spectrum, so as to unveil both embedded eigenvalues and complex plasmonic resonances. The later are analogous to scattering resonances with the local behavior at the corner playing the role of the behavior at infinity. We illustrate in particular the study of Li and Shipman (J. Integral Equations and Appl., 31(4), 2019), which proved the existence of embedded eigenvalues for the Neumann-Poincaré operator for a geometry with reflectional symmetry.