What exactly is a number? We've just been learning about complex numbers in class, and I don't really see why they're called numbers.
Originally, a number used to be a means of counting (natural numbers).
Then we extend these numbers to instances of owing other people money for instance (integers).
After that, we consider fractions when we need to split things like 2 pizzas between three people.
We then (skipping the algebraic numbers for the purposes of practicality), we use the real numbers to describe any length; e.g. the length of a diagonal of a unit square.
But this is when our original definition of a number fails to make sense- when we consider complex numbers, which brings me to my main question: what is a rigorous definition of 'number'?
Wikipedia claims that "A number is a mathematical object used to count, label, and measure", but this definition fails to make sense after we extend $\mathbb{R}$.

Does anyone know of any definition of number that can be generalised to complex numbers as well (and even higher order number systems like the quaternions)?


There is no concrete meaning to the word number. If you don't think about it, then number has no "concrete meaning", and if you ask around people in the street what is a number, they are likely to come up with either example or unclear definitions.
Number is a mathematical notion which represents quantity. And as all quantities go, numbers have some rudimentary arithmetical structures. This means that anything which can be used to measure some sort of quantity is a number. This goes from natural numbers, to rational numbers, to real, complex, ordinal and cardinal numbers.
Each of those measures a mathematical quantity. Note that I said "mathematical", because we may be interested in measuring quantities which have no representation in the physical world. For example, how many elements are in an infinite set. That is the role of cardinal numbers, that is the quantity they measure.

So any system which has rudimentary notions of addition and/or multiplication can be called "numbers". We don't have to have a physical interpretation for the term "number". This includes the complex numbers, the quaternions, octonions and many, many other systems.
