Are there any examples of vector spaces over non-numerical fields? If not, why not? By non-numerical vector spaces I mean vector spaces that do not have as their scalars some sort of easily discernible numerical fields (e.g. complex numbers, functions are usually maps from one numerical space to another, etc.).
Are there any examples of non-numerical vector spaces? If not, why not?
I know that this question asked for something similar, but its accepted answer gave "numerical vector spaces", and more importantly the motivation behind the question was pedagogical.
I on the other hand, am asking in order to determine if there would be any drawbacks to building an algebra that assumes its scalars are all reals, and that complex numbers, quarternions, etc. are built into its higher dimension forms (e.g. geometric algebra/Clifford algebra), as such an algebra might perhaps not be able to capture a useful vector space over a non-numerical field.
 A: The scalars of a vector space just have to be a field. 
The examples $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, etc. which I guess is what you mean by "numeric" are not the only examples of fields.
For instance, there is also plenty of fields of functions: rational functions, meromorphic functions, in general, any quotient field of any however defined ring.
That is, you can just as well form a vector space with the scalars being rational functions instead of "just numbers".
Example:
Consider the space of all pairs $(f,g)$ of rational functions $f,g$ equipped with componentwise addition, i.e. $(f_1,g_1)+(f_2,g_2)=(f_1+f_2,g_1+g_2)$ and the scalar multiplication $f\cdot(g,h)=(fg,fh)$. 
This is a two dimensional vector space over the space of rational functions.
But: 
No matter how "abstractly" you define your field and how you denote its elements, you will always find "numbers" inside.
Consider the following. Let $K$ be a field of characteristic $0$. Let $e$ be the unit in $K$. Then refer to a sum of $n$ times $e$ as $n\cdot e$ ($n$ is a natural number $\ge 1$). 
Since $K$ is a field, given any $n\cdot e$ there exists a multiplicative inverse, which we shall call $\frac{1}{n}\cdot e$ (here we use that $K$ is of characteristic zero).
Likewise, there exists an additive inverse of $n\cdot e$, which we shall call $-n\cdot e$.
Now, consider $n\cdot e\in K$ for some $n\in\mathbb{Z}$ and $\frac{1}{m}\cdot e$ for some $m\in\mathbb{N}$. We denote their product in $K$ by $\frac{n}{m}\cdot e$ (we tacitly named the additive neutral element (that is the "zero") $0\cdot e$).
What have we done here? We constructed an injective field homomorphism
$$\mathbb{Q}\hookrightarrow K$$
sending $\frac{n}{m}$ to $\frac{n}{m}\cdot e$ (you can check readily that it is well-defined and really an injective field homomorphism).
That means: You will always find a copy of $\mathbb{Q}$ in your field. You cannot escape "numbers".
Note: If $\mathrm{char} K=p$ instead, then you find a subfield isomorphic to $\mathbb{Z}/(p)$ by the same procedure.
A: When we want to construct a vector space "containing no numbers", we can proceed by a very simple method:
 1. Chose an Alphabet $\mathfrak A$, i.e. an ordered set of tokens of some sort. examples:
$$\{A,B,C,\ldots,Z,AA,AB,\ldots,ZZ\}$$
$$\mathfrak A =\{\text{car, boat, ship, elephant}\}$$
 2. Chose a prime number $p$, f.ex. $p=2$
 3. if $\phi$ is an enumeration of our alphabet, we can define a Field by
$$\mathcal A :=\{\phi(i) \in \mathfrak A | 0 \leq i<p\}$$
$$+: \mathcal A\times\mathcal A\to \mathcal A, \qquad a+b \mapsto \phi(\phi^{-1}(a)+\phi^{-1}(b))$$
$$\cdot: \mathcal A\times\mathcal A\to\mathcal A, \qquad a\cdot b \mapsto \phi(\phi^{-1}(a) \cdot \phi^{-1}(b))$$
 4. chose a dimension $n\in\mathbb N$ and consider $\mathcal A^n$ as your vector space (f.ex. $n=2$)

To elaborate on an example, we chose $\def\t#1{\text{#1}}$
$$\begin{align*}
\mathfrak A & := \{\text{car, boat, ship, elephant}\} \\
p & = 3\\
\Rightarrow \mathcal A & = \{\text{car,boat,ship}\} \\
\text{multiplication:} &
\begin{pmatrix}\cdot&\t{car}&\t{boat}&\t{ship}\\
\t{car}&\t{car}&\t{car}&\t{car}\\
\t{boat}&\t{car}&\t{boat}&\t{ship}\\
\t{ship}&\t{car}&\t{ship}&\t{boat}\end{pmatrix}\\
\t{addition:} & \begin{pmatrix} +&\t{car}&\t{boat}&\t{ship}\\
\t{car}&\t{car}&\t{boat}&\t{ship}\\
\t{boat}&\t{boat}&\t{ship}&\t{car}\\
\t{ship}&\t{ship}&\t{car}&\t{boat}
\end{pmatrix} \\
n &=2 \\
0\text{-vector} & = \begin{pmatrix}\t{car}\\\t{car}\\\t{car}\end{pmatrix}\\
\ldots
\end{align*}$$
A: Example of a vector space over a non-numerical field:
consider the set $k\left[ x \right] $ of polynomials with coefficients in the field $k$. Then $k\left[ x \right] $ is a vector spaces over $k$, with the usual operations of polynomials. 
This is true for every field. Then, if $k$ is a 'non-numerical field' then $k\left[ x \right] $ is the vector space that you are looking for. Now can you give an example of a 'non-numerical field'?
