Better I think would be to say "the result in column i and row j is the sum of product of elements in column i of the left cracovian and column j of the right cracovian".
And even by this definition the example given doesn't seem to track (and the strangeness of sometimes saying "+" and sometimes not, and having both "0" and "-0" in the example is bananas!):
{ 3 2 } { 1 -4 } = { 5 -2 }
{ -1 0 } { -2 3 } = { 0 2 }
3 * 1 + -1 * -2 == 5 -- check
3 * -4 + -1 * 3 == -15 -- what?
2 * 1 + 0 * -2 == 2 (okay, but shouldn't this be in the lower left, column 1 dotted with column 2?)
2 * -4 + 0 * 3 = -8 (now I'm really missing something)
A unit quaternion represents a rotation in R³.
A quaternion represents the quotient of two vectors in R³. That's what Hamilton had in mind.