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.
An inner product is a symmetric positive definite bilinear form. In general fields, you'll happily be able to satisfy symmetric bilinear, but you'll struggle with positive definite: over a field of nonzero characteristic, you will not even be able to make sense of $\langle x, x \rangle \ge 0$, much less find a form for which it holds, since there is no ordering compatible with the field. Note that there is also no ordering on the complex numbers that is compatible with the field, but there is the subfield $\mathbb R$ which can be ordered, and so conjugate symmetry rescues us by ensuring $\langle x, x \rangle$ falls inside there, but fields of characteristic $p$ must contain a subfield $\mathbb F_p$ which is already not orderable.
Over fields of characteristic zero, like the rationals, you can find a reasonable inner product, but it's not as useful as you'd expect. For example, you can't get an orthonormal basis from a given basis, because you can't do square roots: you can find the norm squared, but not the norm itself. You could go all the way to the algebraics, or just as far as all the square roots, but at this stage I think you gain very little generality over just using $\mathbb R$ in the first place, and if you find you want completeness at any point, you'll be forced into the reals anyway.
I think I have too found the asymmetry between complex and real inner products to be frustrating. It's possible that there's a unifying theory that I'm unaware of, but I don't think it's unreasonable to view them as basically separate (if highly similar) entities, albeit entities that embed one in the other in a neat way. Essentially, $\mathbb R$ is special: it is, after all, the unique complete totally ordered Archimedean field, so it's not that surprising that we should pay it specific attention.
Best Answer
This is a rather approximative overview of what generalizations can be explored in an early course of linear algebra.
The short answer is that all that does not use the fact that $\Bbb R$ is ordered, $\Bbb C$ has a norm, or that $\Bbb C=\Bbb R[i]=\overline{\Bbb R}$ carries on identically to all fields and it can, in principle and in point of fact, be taught directly as "linear algebra", instead of "$\Bbb R$-or-$\Bbb C$ linear algebra". More specifically
All the things that are genuinely linear, like basis, matricial representations for finite dimensional spaces, dual and bi-dual, Gaussian elimination, determinants, Rouché-Capelli theorem carry on verbatim or with very obvious adjustments.
The results around Jordan normal form stay unchanged for algebraically closed fields. Phenomena like "real Jordan normal form", though, use heavily the fact that $\dim_{\Bbb R}\Bbb C=2$, and need to be heavily amended to be generalized to other extensions (which are almost always of infinite degree) in an interesting way.
The "theory of real inner products on finitely dimensional spaces" is generalized by the theory of quadratic forms, and it is interesting even as part of an early course. It studies the symmetric and bilinear maps $\phi:k^n\times k^n\to k$. There is a generalized notion of orthogonality, of adjoint, of degenerate quadratic forms, of orthogonal maps (sometimes called isometries). The main differences revolve around the fact that:
$\Bbb R$ is ordered, and so there is a notion of sign and positive definiteness that can be used to control/distinguish a lot of things. For a general field, the only thing that can be controlled is the presence of vectors $v$ such that $\phi(v,v)=0$ (isotropic vectors) and/or such that $\phi(v,w)=0$ for all $w$ (orthogonal to the whole space). This reflects on terminology and choice of "canonical forms". If you want to quickly gage the flavour of it, have a look at these results by Witt.
Fields of characteristic $2$, and $\Bbb F_2$ especially, need (if any) a separate treatment. The issue is that, in fields where $1+1\ne0$, there is a bijective correspondence between symmetric bilinear maps and homogeneous polynomial functions of degree $2$ - i.e., maps $q:k^n\to k$ that can be written as $q(v)=\sum_{i,j}q_{ij}v_iv_j$ for some constants $q_{ij}$. This correspondence is established by calling $Q_{\phi}(v)=\phi(v,v)$, and $\Phi_q(v,w)=\frac{q(v+w)-q(v)-q(w)}{2}$. It's straight-forward to verify that $\Phi_{q_\phi}=\phi$ and $Q_{\Phi_q}=q$. You can't divide by $2$ when $1+1=0$, and it turns out that the map $\phi\mapsto Q_\phi$ is not injective in characteristic $2$.
However, you may want to look into an actual textbook for further detail on (3); "Introduction To Quadratic Forms Over Fields" by Lam (or its earlier, more famous version "Algebraic Theory of Quadratic Forms") is something that you may find in your local library. It doesn't quite cover what happens in characteristic $2$, though.