Why is a field $\mathbb{F}$ "always" chosen as $\mathbb{R}, \mathbb{C}$? In my experience I have never encounter a problem where a field $\mathbb{F}$ is not $\mathbb{R}$ or $\mathbb{C}$. I know that $\mathbb{R}, \mathbb{C}$ are complete in Euclidan metric and perhaps that is why we almost everytime choose $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$ because it has nice structure. I can therefore imagine that some problem would be much harder if you would pick an arbitrary field. But what reasons are there? 
I'm taking functional analysis right now, and it was in that class that the question came to me.
 A: This depends strongly on the area you want to work in. Especially in Analysis and areas building on Analysis like ODE, functional analysis etc $\mathbb{R}$ and $\mathbb{C}$ are natural choices for the fields to use since Analysis usually strongly depends on completeness. 
A: There's a lot of fields out there, and the field is not "always" chosen as $\mathbb{R}$ or $\mathbb{C}$.
Number theory, specifically algebraic number theory, uses finite fields, as well as "number fields" which are those fields that have a subfield isomorphic to the field $\mathbb{Q}$ of rational numbers and that are finite dimensional over $\mathbb{Q}$. At more advance levels of number theory, there are two other types of fields that come into play: the algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$, in which all the number fields live; and the fields $p$-adic numbers $\mathbb{Q}_p$ and other fields derived from them.
Algebraic geometry studies "function fields" which are obtained by taking a polynomial ring like $\mathbb{R}[x]$, which denotes the ring of polynomials in the variable $x$ having real coefficients, and forming its "fraction field" $\mathbb{R}(x)$ which is the field of rational functions written as a fraction with numerator and denonimator both taken from $\mathbb{R}[x]$. 
Here's a nice place where algebra and analysis overlap: the field $\mathbb{C}(x)$ of rational functions with complex coefficients is of key importance in studying the Riemann sphere $\mathbb{C}^*$. An important fact here is that $\mathbb{C}(x)$ is naturally isomorphic to the field of meromorphic functions on $\mathbb{C}^*$.
