Why Do We Only Take Norms Over Real/Complex Numbers? By definition, norms are defined over some $\mathbb{R}$ or $\mathbb{C}$ vector space. Why do we only restrict ourselves to these fields when other fields give rise to interesting objects as well? (e.g. p-adic evaluation)
Is it a historic reason or because other fields would give properties so different that we‘d rather not also associate the term “norm“ with it? If so, then I assume that $\mathbb{R}$ and $\mathbb{C}$ are similar enough to make those the two fields that give rise to norms?
 A: I don't think this is true. People who work in $p$-adic analysis often work with with the quantity $|\vec{x}| = \max_{1 \leq i \leq n} (|x_i|_p)$ for $\vec{x} \in \mathbb{Q}_p^n$, with $|\ |_p$ the $p$-adic norm. This is a norm in the sense that $|\vec{x}+\vec{y}| \leq \max(|\vec{x}|, |\vec{y}|)$, $|c \vec{x}| \leq |c|_p |\vec{x}|$ and $|\vec{x}|=0$ if and only if $\vec{x} = \vec{0}$. The metric induced by this norm on $\mathbb{Q}_p^n$ gives the standard product topology.
The group of matrices preserving this norm is a useful group: It is the matrices $g$ for which both $g$ and $g^{-1}$ have entries in $\mathbb{Z}_p$. It is usually denoted $GL_n(\mathbb{Z}_p)$, and plays the analogue of the orthogonal group. Indeed, Smith normal form for the PID $\mathbb{Z}_p$ says that every matrix in $GL_n(\mathbb{Q}_p)$ can be factored as $U \Sigma V$ where $U$ and $V$ are in $GL_n(\mathbb{Z}_p)$ and $\Sigma$ is diagonal with entries powers of $p$; it is valuable to think of this as a non-Archimedean analogue of singular value decomposition.
I learned this perspective from Chapter 4 of Kiran Kedlaya's book "$p$-adic differential equations" and I have seen plenty of other $p$-adic papers use it since.
I just looked at the OP's bio, and it looks like they are a young undergraduate. So the reason they haven't seen this might just be that linear algebra books written for undergraduates don't assume the reader knows what the $p$-adics are.
A: A norm on a space $V$ also induces a metric on $V$ (more or less by definition), and a metric is by definition a map from $V \times V \rightarrow \mathbb{R}$. So, the norm should map the elements of $V$ to $\mathbb{R}$, and we require that $\left\|\alpha x\right\| = |\alpha| \left\|x\right\|$ for norm properties. So, $|\alpha|$ must also be a real number for any $\alpha$ in the field which $V$ is over. This pretty much limits the choice of field to $\mathbb{R}$ and $\mathbb{C}$.
Of course, this answer pretty much just pushes the question to why we want metrics to map to $\mathbb{R}$...
A: I think that if you want to work with norms on vector spaces over fields in general, then you have to use the concept of valuation. 
Valued field:
Let $K$ be a field with valuation $|\cdot|:K\to\mathbb{R}$. This is, for all $x,y\in K$, $|\cdot|$ satisfies:


*

*$|x|\geq0$,

*$|x|=0$ iff $x=0$,

*$|x+y|\leq|x|+|y|$,

*$|xy|=|x||y|$.


The set $|K|:=\{|x|:x\in K-\{0\}\}$ is a multiplicative subgroup of $(0,+\infty)$ called the value group of $|\cdot|$. The valuation is called trivial, discrete or dense accordingly as its value group is $\{1\}$, a discrete subset of $(0,+\infty)$ or a dense subset of $(0,+\infty)$. For example, the usual valuations in $\mathbb{R}$ and $\mathbb{C}$ are dense valuations.
Norm: Let $(K,|\cdot|)$ be a valued field and $X$ be a vector space over $(K,|\cdot|)$. A function $p:X\to \mathbb{R}$ is a norm iff for each $a,b\in X$ and each $k\in K$, it satisfies:


*

*$p(a)\geq0$ and $p(a)=0$ iff $a=0_X$,

*$p(ka)=|k|p(a)$,

*$p(a+b)\leq p(a)+p(b)$
There is a whole research area in which arbitrary valued fields are considered and these fields are not necessarily ordered fields. It is called non-Archimedean Functional Analysis. A comprehensive starting point to read about normed spaces in this context is the book: Non-Archimedean Functional Analysis - [A.C.M. van Rooij] - Dekker New York (1978). 
For the study of more advanced stuff, like locally convex spaces over valued fields I recommend the book: Locally Convex Spaces over non-Arquimedean Valued Fields - [C.Perez-Garcia,W.H.Schikhof] - Cambridge Studies in Advanced Mathematics (2010).
Now if you wonder whether the concept of valuation can be generalized, the answer is yes. On a field $K$ you can take a map $|\cdot|:K\mapsto G\cup\{0\}$ satisfying


*

*$|x|\geq0$,

*$|x|=0$ iff $x=0$,

*$|x+y|\leq max\{|x|,|y|\}$,

*$|xy|=|x||y|$.


where $G$ is an arbitrary multiplicative ordered group and $0$ is an element such that $0<g$ for all $g\in G$. In this new setting, a norm can take values in an ordered set $Y$ in which $G$ acts making of $Y$ a $G$-module. 
For an introdution in this area I recommend the paper:
Banach spaces over fields with a infinite rank valuation, In J. Kakol, N. De Grande-De Kimpe, and C. Perez-Garcia, editors, p-adic Functional Analysis, volume 207 of Lecture Notes in Pure and Appl. Math., pages 233-293. Marcel Dekker - [H.Ochsenius A., W.H.Schikhof] - 1999
After that see: Norm Hilbert spaces over Krull valued fields - [H. Ochsenius, W.H. Schikhof] - Indagationes Mathematicae, Elsevier - 2006
