Quaternions
can be a fun topic,
esp. to explain where the stupid shit in vector calc comes from
(over-enthusiastic physicists decided to consider $\bR^3$ as imaginary
quaternions (hence the basis vectors $i,j,k$, and the emphasis on the
cross product). This was a bad idea.)

multiplication:
it's a good idea to draw an ijk circle (order matters!),
as a mneumonic for
$ij=k, jk=i, ki=j$
[which is equivalent to $ijk=-1$, in the presence of
 $i^2=j^2=k^2=-1$]
...this leads naturally to octonions

[formally, $i^2=j^2=k^2=ijk=-1$ is all,
 but it's a lil' confusing b/c it conflates the squares and the
 line]

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There's of course Conway's book on it.
