Is 'det' a single function? I should mention this up front: this is a question about semantics. It is extremely formal, and a bit pedantic. Most people won't care.

Is det, i.e the determinant function, a single one? Or is it multiple functions, one for each field?

Let $\mbox{Mat}(K)$ denote the set of square matrices over the field $K$ (This is nonstandard notation). Then $\det(x) \in K$ for $x\in \mbox{Mat}(K)$. So we can view $\det$ in this context as a function $\det: \mbox{Mat}(K) \rightarrow K$. If you take this view, then you are forced to acknowledge that 'det' actually refers to infinitely many functions, one for each field $K$, and that which 'det' we are referring to in a particular expression depends entirely on context.

I should note that here, we take the view that a function MUST possess a domain and codomain, that it is IS NOT just a collection of (input, output) pairs with some sort of implicit domain and codomain.

On the other hand, we may view $\det$ as a single function, taking in ANY square matrix, and returning... something. This is fine, but what would the codomain be? It would have to consist of... everything, and that is not a good codomain for a single function object. Ideally, the codomain would be a bit more expressive. However, this, formally, is not a problem. 

So, which is it? The former or the latter?
The meaning that mathematicians usually implicitly have in mind is that there are lots of separate functions written "$\det$".  Specifically, for each field $K$ and each $n\in\mathbb{N}$, there is a separate function $\det:M_n(K)\to K$ (which to be completely pedantic perhaps should be written $\det_{n,K}$, but that quickly gets unwieldy so no one does it).  Note here that $M_n(K)$ is the standard notation for the set of $n\times n$ matrices with entries in $K$.

There are also many other functions that are commonly written as "$\det$".  For instance, it is very common to restrict one's attention to invertible matrices and write $\det:GL_n(K)\to K^\times$, where $GL_n(K)$ is the set of invertible $n\times n$ matrices and $K^\times$ is the set of nonzero elements of $K$.  This is of course not literally the same function as the function $\det:M_n(K)\to K$, but no one ever bothers to distinguish it notationally.

I would add that absolutely none of this is special to "$\det$".  There are oodles and oodles of similarly context-dependent notations in math; for instance, the symbol $+$ can refer to a vast variety of different binary operations, with the specific operation in any given instance usually left to be inferred from context.  Mathematical notation is meant to be understood by humans, and humans are very good at resolving ambiguity from context (and bad at processing overly cluttered but completely unambiguous notation).
What about define it as a function with following domain and codomain 
$$
\mathrm{det}: \bigcup_{n=1}^{\infty}\ \bigcup_{K:\ K\ \text{is a field}} M_{n\times n}(K)\to \bigcup_{K:\ K\ \text{is a field}} K
$$