Vector spaces over fields other than $\mathbb{R}$, $\mathbb{C}$, & $\mathbb{F}_{p^m}$ Are there any useful applications of vector spaces over fields other than $\mathbb{R}$, $\mathbb{C}$, and finite fields?  For example, are vector spaces over the rationals $\mathbb{Q}$ used for anything?  How about fields larger than $\mathbb{C}$ or $\mathbb{R}$?
 A: More generally, why stop at fields? You can have "vector spaces" over rings, though we usually call them modules over rings. These are ubiquitous in mathematics: for instance the Serre-Swan theorem identifies certain modules with vector bundles over a space.
If you're looking for applications of vector spaces over fields (other than $\mathbb{R}$ or $\mathbb{C}$), specifically, then these often come up in algebraic topology. For instance, this paper discusses the rational cohomology of Lie groups, and also cohomology with coefficients in various finite fields. If you google "rational homology" or "rational cohomology" you'll find plenty more examples.
A: The classic Galois Theory by Emil Artin uses linear algebra as a foundation. 
The number fields that appear in algebraic number theory are by definition finite-dimensional vector spaces over the rationals.
Finite fields are used in error detection and correction.
A: Also in algebraic geometry, the tangent and cotangent spaces at a point of a variety are vectors spaces over the residue field at the point.
A: Essentially computer graphics (so think Pixar, Disney, computer games) all use rational point geometry (ie vector spaces over $Q$).
The same goes for 3d printing software.
Is this the sort of thing you meant?
