For me, back in school days (or was it university?) it was a revelation when I came across quaternions. It suddenly clicked. I finally understood that there was nothing special about complex numbers (despite their special names and weird look). It was just an extension and a very intuitive one!
I finally understood that real numbers are the same thing: tuples. They just happen to have exactly one element, hence we omit parens and everything else and just write that element (number)! Complex numbers have 2 elements (real and imaginary). Quaternions - four. And so on.