The eigenvalues of a 2d rotation matrix are complex numbers. There's no point in trying to visualise the corresponding eigenvectors. You'll need 4 dimensions for that. (The 4 dimensions come from the fact that the eigenvectors of a 2d rotation matrix are elements of ℂ^2, which is topologically equivalent to (ℝ^2)^2, which is equiv to ℝ^4).

Personally, my "intuition" is based on analogies with non-complex eigenvectors, and experience solving eigenvector problems algebraically without using pictures.

Also, your 3d rotation example actually has three eigenvectors, two of which are complex. You've only found the one that's real.