Let's assume the basic nouns of our language to describe the physical world are the members of Lie groups. Okay, this is a pompous-sounding statement and somewhat arbitrary, but my justification is that these objects describe all the continuous symmetries there can be, and almost every clarification of physics using mathematics is done either (1) by viewing a mathematical object from a different standpoint (unification of hitherto seemingly unrelated concepts) or (2) by exploiting symmetries to reduce or get rid of the redundant complexity in a statement. In our continuous manifold descriptions of the physical World, these symmetries are all continuous. So, somewhere in that list of symmetries, we meet $U(1)$, $SU(2)$, $SO(3)$, $U(N)$ and so forth. So we would needfully be doing calculations and simplifcations with these objects when we exploit symmetries of a problem. Whether or not we choose to single out an object like:
$$\left(\begin{array}{cc}0&-1\\1&0\end{array}\right)\in U(1), SU(2), SO(3), U(N) \cdots$$
and give it a special symbol $i$ where $i^2=-1$ is a "matter of taste", so in this sense the use of complex numbers is not essential. Nonetheless, we would needfully still meet this object and ones like it and would have to handle statements involving such objects when describing physics in a continuous manifold - there's no way around this as it belongs to any full description of symmetries of the World. So in this sense, complex numbers, quaternions, octonions and so forth are all there and essential in such description. Notice that complex numbers and their algebra are wonted to almost everyone in physics, quaternions to somewhat fewer physicists and octonions not really to that many. This is simply related to how often the relevant symmetry calculations come up: almost any interesting continuous symmetry involves Lie group objects for which $i^2=-1$ and so we single these out and commit all the rules of their algebra to stop ourselves going outright spare and committed to lunatic asylums writing out their full Lie theoretical representations all the time. Singling out quaternions and doing the same saves some work, but not so much, because quaternions come up in fewer symmetries. By the time we get to octonions, the symmetries wherein they come up are quite seldom, so not that many of us are very adept with their special algebra (me included): we can do the full matrix / Lie calculations without too much pain because we don't do them that often, so we don't notice their octonionhood so readily.
Footnote: One can take "Lie Group members" and "Continuous Symmetries" to be the same by dint of:
- The solution to Hilbert's fifth problem by Montgomery, Gleason and Zippin i.e. we don't need the concept of manifold nor the concept of analyticity ($C^\omega$) - these "build themselves" from the basic idea of a continuous topological group;
- The classification of all Lie algebras by Wilhelm Killing (whose saw that he could do it, but botched the proof a little) and the great Elie Cartan - so we know what all continuous symmetries look like. Once we have classified all Lie algebras, we can find all possible Lie groups, since every Lie group has a Lie algebra, every Lie algebra can be exponentiated into a Lie group (e.g. through the matrix exponential, since every Lie algebra can be represented as a matrix Lie algebra (Ado's theorem)) and the (global-topological) relationships between Lie groups that have the same Lie algebra is also known.
