I'm curious if people study analysis while using fields that are not $\mathbb{R}$. I remember seeing a post about doing analysis on $\mathbb{Q}$, but $\mathbb{Q}$ is not complete! Mostly I'm interested in applying functional analysis with differing underlying fields. The fields I have in mind are like field extensions of a given polynomial, or a given set of polynomials. I'm unsure if one can construct a complete field in this way (without constructing the reals). This seems to dip into algebra, analysis and topology so for me, it's probably a bit to advanced to actually make any progress with alone.
Has anyone done analysis in this sort of way? Are there any applications whatsoever (am I being incredibly silly considering this)?
If you want to do analysis over some field $\mathbb{K}$ you need some kind of absolute value on it (sometimes called a valuation). There are only two types of valuations: archimedean and non-archimedean. And this claim is not a tautology, see lemma 1.2 in Local fields. J. W. S. Cassels. Simply speaking archimedean valuations are good and non-archimedean are weird.
By Ostrowski's theorem, there are only two archimedean valued fields: $\mathbb{R}$ and $\mathbb{C}$. So this part of analysis is very well understood. The remaining part is usually called non-archimedean. It is not so well studied but, there is a lot of literature on this question already. The main example of a complete non-archimedean field is $\mathbb{Q}_p$. It is so important that non-archimedean analysis is sometimes called the $p$-adic analysis.
Long story short, a lot of things in non-archimedean analysis is significantly trivializes or not true in general. Examples of drawbacks:
non-archimedean fields have no natural $<$ relation. As a consequence, you cannot build a good theory of integration.
non-archimedean fields have no natural involution (like $\mathbb{C}$), so you cannot make a bunch of things in complex analysis. For the same reason, you cannot build a decent theory of involutive algebras and $C^*$-algebras.
non-archimedean fields can be non-separable and not even locally compact.
Liouville's theorem is true only for algebraically closed non-archimedean fields.
A non-archimedean field must be spherically complete in order to have Hahn-Banach theorem at hand. So for non-spherically complete fields, you cannot do a lot of things in functional analysis.
Non-archimedean fields do not always possess a Haar measure on non-discrete locally compact groups, so in most interesting cases you cannot do abstract harmonic analysis.
Non-archimedean fields have no analog of Gelfan-Mazur's theorem, so a big part of the spectral theory is unavailable.
Non-archimedean local fields are totally disconnected, so continuity theory evolves in a different manner.
Non-archimedean fields are so restrictive, that most non-archimedean Banach spaces are linearly topologically isomorphic to $c_0(S)$ for some index set $S$. Therefore, locally convex spaces are of much more interest than Banach spaces.
Of course, these drawbacks make non-archimedean analysis interesting in some sense, because a researcher has the challenge to extend the well-known theories to the cases where all standard tools do not work.
If you are mainly interested in functional analysis I recommend taking a look at this discussion. For the first reading I recommend Non-archimedean functional analysis A. C. M. van Rooij and for the more modern treatment Nonarchimedean functional analysis P. Schneider and after that Locally convex spaces over non-archimedean valued fields. C. Perez-Garcia, W. H. Schikhof.
If you don't have enough time to read all that books take a look at this presentation Introduction to non-archimedean functional analysis. W. Sliwa