eigenvalues over C vs R
Eigenvalues are roots of the characteristic polynomial, and the field you allow those roots to live in changes the answer dramatically. Over the real numbers a polynomial may have no roots at all (think x^2 + 1), so a perfectly ordinary real matrix can have no real eigenvalues. Over the complex numbers every polynomial of degree n splits completely — this is why C is the natural home for eigentheory.
The cleanest example is a rotation. The matrix R = [cos t, -sin t; sin t, cos t] rotates the plane by angle t and (for t not a multiple of pi) fixes no real direction, so it has no real eigenvectors. Its characteristic polynomial lambda^2 - 2 cos t lambda + 1 has complex roots e^{i t} and e^{-i t}. The geometry of pure rotation is encoded entirely in those complex eigenvalues.
Real matrices keep one piece of nice structure: their complex eigenvalues come in conjugate pairs, lambda and its conjugate, with conjugate eigenvectors. This is why a real matrix that is not diagonalizable over R can still be brought to a real block form with 2x2 rotation-scaling blocks (the real Jordan / real canonical form), capturing each complex pair without leaving the reals.
A real rotation has no real eigenvectors but a clean pair of complex eigenvalues on the unit circle.
The Fundamental Theorem of Algebra is what makes C the natural field: it guarantees the characteristic polynomial splits, so over C every n-by-n matrix has n eigenvalues counted with multiplicity, and every such matrix is at least triangularizable.