Also, in Hamilton's time they didn't have the view of mathematics as axiom systems that could be played with. Math was a way of finding Truth, so the idea of making up multiplication rules at your convenience probably seemed like an extremely non-obvious move.

The discovery of complex numbers and quaternions probably played a big part in getting people to question what math is, leading to Hilbert's program to study Foundations etc. Hamilton's story is a nice, rare single instance we can point to, symbolizing this discovery.