It is said that if we multiply a real eigenvector by any real number k and hence we get another vector lying on the same line, the new one is also an eigenvector.
Let us consider now for example a rotation matrix, whose complex eigenvectors are (1,i) and (1,-i). The eigenvalue associated to the first one is $e^{i\theta}$. We can now consider multiplying by this eigenvalue but with any possible value for the angle $\theta$ or any possible value for a pre-factor $r$ indicating magnitude. Would the resulting vectors be also complex eigenvectors? Can you say that this way you “span” the whole complex plane in the relevant direction, here the positive or counterclockwise direction?