In my experience, when working over a division ring $D$, the main thing you have
to be careful of is the distinction between $D$ and $D^{op}$.
E.g. if $F$ is a field, then $End_F(F) = F$ ($F$ is the ring of $F$-linear endomorphisms of itself, just via multiplication), and hence $End(F^n) = M_n(F)$;
and this latter isomorphism is what links matrices and the theory of linear transformations.
But, for a general division ring $D$, the action of $D$ by left multiplication on itself is not $D$-linear, if $D$ is not commutative. Instead, the action of $D^{op}$ on $D$ via right multiplication is $D$-linear, and so we find that
$End_D(D) = D^{op}$, and hence that $End_D(D^n) = M_n(D^{op}).$
As for examples of division algebras, they come from fields with non-trivial Brauer groups, although this may not help particularly with concrete examples.
A standard way to construct examples of central simple algebra over a field $F$ is via a crossed product. (Unfortunately, there does not seem to be a wikipedia entry on this topic.)
What you do is you take an element $a\in F^{\times}/(F^{\times})^n$, and
a cyclic extension $K/F$, with Galois group generated by an element $\sigma$
of order $n$, and then define a degree $n^2$ central simple algebra $A$ over $F$
as follows:
$A$ is obtained from $K$ by adjoining a non-commuting, non-zero element $x$,
which satisfies the conditions
- $x k x^{-1} = \sigma(k)$ for all $k \in K$, and
- $x^n = a$.
This will sometimes produce division algebras.
E.g. if we take $F = \mathbb R$, $K = \mathbb C$, $a = -1$, and $\sigma =$ complex conjugation, then $A$ will be $\mathbb H$, the Hamilton quaternions.
E.g. if we take $F = \mathbb Q_p$ (the $p$-adic numbers for some prime $p$),
we take $K =$ the unique unramified extension of $\mathbb Q_p$ of degree $n$,
take $\sigma$ to be the Frobenius automorphism of $K$,
and take $a = p^i$ for some $i \in \{1,\ldots,n-1\}$ coprime to $n$,
then we get a central simple division algebra over $\mathbb Q_p$, which is called the division algebra over $\mathbb Q_p$ of invariant $i/n$ (or perhaps $-i/n$, depending on your conventions).
E.g. if we take $F = \mathbb Q$, $K =$ the unique cubic subextension of $\mathbb Q$ contained in $\mathbb Q(\zeta_7)$, and $a = 2$, then we will get
a central simple division algebra of degree $9$ over $\mathbb Q$.
(To see that it is really a division algebra, one can extend scalars to $\mathbb Q_2$, where it becomes a special case of the preceding construction.)
See Jyrki Lahtonen's answer to this question, as well as Jyrki's answer here, for some more detailed examples of this construction. (Note that a key condition for getting a division algebra is that the element $a$ not be norm from the extension $K$.)
Added: As the OP remarks in a comment below, it doesn't seem to be so easy to find non-commutative division rings. Firstly, perhaps this shouldn't be so surprising, since there was quite a gap (centuries!) between the discovery of complex numbers and Hamilton's discovery of quaternions, suggesting that the latter are not so easily found.
Secondly, one easy way to make interesting but tractable non-commutative rings is to form group rings of non-commutative finite groups, and if you do this over e.g. $\mathbb Q$, you can find interesting division rings inside them. The one problem with this is that a group ring of a non-trivial group is never itself a division ring; you need to use Artin--Wedderburn theory to break it up into a product of matrix rings over division rings, and so the interesting division rings that arise in this way lie a little below the surface.
One of the ingredients of a vector space is a definition of scalar multiplication; but you need to know what field the scalars are in! A vector space $V$ over $\mathbb{F}$ has as part of its data a map (scalar multiplication) $\mathbb{F}\times V\to V$. The same set $V$ could be given a different vector space structure with a multiplication map $\mathbb{K}\times V$; this is most common when $\mathbb{K}\subset\mathbb{F}$, and the second map is simply a restriction of the first one.
For example, $\mathbb{C}$ is a vector space over $\mathbb{C}$ as you mention in the question, but it is also a vector space over $\mathbb{R}$, since you can simply restrict scalar multiplication to real scalars. These two structures are quite different; for example, the single element $1$ is a basis of $\mathbb{C}$ over $\mathbb{C}$, since every complex number is of the form $z\cdot 1$ for some unique $z\in\mathbb{C}$. Over $\mathbb{R}$, one possible basis is $1,i$, since every complex number is of the form $a+bi$ for a unique pair $(a,b)\in\mathbb{R}^2$. Notice that the dimension of $\mathbb{C}$ changed depending on whether we gave it the structure of a vector space over $\mathbb{C}$ or over $\mathbb{R}$.
Best Answer
You could define an inner product on an ordered field, as you need to satisfy the positive-definiteness axiom. However, without a suitable order on the field, this axiom is meaningless. In order to generalise the conjugation in the definition of a hermitian inner product, you can introduce an involution on a field. As long as you can introduce an order and an involution, you should be able to generalise the definition easily enough.
As an aside, it's worth noting that when you do away with the positive-definiteness axiom, what you have is a symmetric bilinear form, which you can define on most (all? I'm not 100%!) fields.