I'm reading a paper about Hamilton's discovery of quaternions and it explains why he failed in his 'theory of triplets' where he tried to make a vector with $3$ dimensions, as an analogy to the complex field, where we can see a number as a $2$ dimensional vector.
In this paper, he explains why is impossible to create a field with $3$ components, that is an extension of the complex field (in other words, it respects addition, and multiplication in the same way...). Here it is:
As you can see, it goes through all the possibilities and proves that it is impossible.
The paper, however, does not explain why $j^2=-1$. It could be anything! Why $-1$?
The article itself is pretty intuitive, but this aspect kills me.
Later, in the article, it says that we should instead consider a 4th component called $k$, such that $k^2=-1$ (also, $i$ and $j$ too).
EDIT: this paper turned out to be extremely helpful