This is more a conceptual question than any other kind. As far as I know, one can define matrices over arbitrary fields, and so do linear algebra in different settings than in the typical freshman-year course.
Now, how does the concept of eigenvalues translate when doing so? Of course, a matrix need not have any eigenvalues in a given field, that I know. But do the eigenvalues need to be numbers?
There are examples of fields such as that of the rational functions. If we have a matrix over that field, can we have rational functions as eigenvalues?
Of course. The definition of an eigenvalue does not require that the field in question is that of the real or complex numbers. In fact, it doesn’t even need to be a matrix. All you need is a vector space $V$ over a field $F$, and a linear mapping $$L: V\to V.$$
Then, $\lambda\in F$ is an eigenvalue of $L$ if and only if there exists a nonzero element $v\in V$ such that $L(v)=\lambda v$.