In 1835, at the age of 30, Hamilton had discovered how to treat complex numbers as pairs of real numbers. Fascinated by the relation between complex numbers and 2-dimensional geometry, he tried for many years to invent a bigger algebra that would play a similar role in 3-dimensional geometry. In modern language, it seems he was looking for a 3-dimensional normed division algebra. His quest built to its climax in October 1843. He later wrote to his son:
Every morning in the early part of the above-cited month, on my coming down to breakfast, your (then) little brother William Edwin, and yourself, used to ask me: "Well, Papa, can you multiply triplets?" Whereto I was always obliged to reply, with a sad shake of the head: `No, I can only add and subtract them".The problem was that there exists no 3-dimensional normed division algebra. He really needed a 4-dimensional algebra.
That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i,j,k; exactly such as I have used them ever since.And in a famous act of mathematical vandalism, he carved these equations into the stone of the Brougham Bridge:
He spent the rest of his life working on quaternions.