f[k_] = {{0.001 - 2 I k + 0.001 k^2, -0.001, 0.001, -0.001}, {-1,
0.5 - I k + 0.001 k^2, 0.501, -0.001}, {0.001, -0.001,
0.001 + 2 I k + 0.001 k^2, -0.001}, {0.501, -0.001, -1,
0.5 + I k + 0.001 k^2}};

r[k_] = Eigenvalues[f[k]][[3]];

Plot[Re[r[k]], {k, -1, 1}]
Plot[Im[r[k]], {k, -1, 1}]