The linear algebra course that I took was fairly consistent about assuming that the scalar field is either the reals or the complex numbers. The theory about linear maps, basis, their matrices, eigenvalues and eigenvectors, trace and determinant clearly generalize to a general field without any changes. Similarly, the Jordan canonical form only seems to require algebraic closedness of a field.
The definition of an inner-product seems to explicitly require either the reals or the complex numbers, but even then one should be able to replace it by a bilinear pairing $V\times V\to \mathbb{F}$, where $\mathbb{F}$ is our field. However, why do we then want conjugate symmetry in the complex case? Do we need something similar for fields which have a "similar" automorphism? Is there a precise way to formalize this?
My questions is essentially the following: How can we generalize spectral theorems to general fields? What would the results look like and what do we need to assume? What's the right way to generalize inner-product spaces and what can we translate unchanged from the setting of an undergraduate linear algebra class?