I understand that in representation theory we try to understand a group $G$ by studying the homomorphisms $\rho\ \colon G \to $ GL$(V)$ where $V$ is a vector space over some field. I believe complex numbers, adic numbers and finite fields are popular because we already understand the linear groups over these fields (though I really only have first-hand experience with the complex numbers). Suppose $G$ was simply "too large" to be a matrix group over $\mathbb C$ say. This can occur since the cardinality of GL$_n(\mathbb C)$ is equal to that of $\mathbb C$. If the cardinality of the group is larger than $\aleph _1$ we cannot hope to get a faithful (injective) representation.
What would be the better idea then, to let $V$ be an infinite-dimensional vector-space, or to use some field of higher cardinality? I imagine the first is the answer, since we also know a lot about infinite dimensional complex vector spaces, from Functional Analysis. But if you had to choose a larger field for some reason while keeping the vector space dimension finite, which field would be a good choice?