I don't pretend to know anything much about the Fundamental Theorem of Algebra (FTA), but I do know what it states: for any polynomial with degree $n$, there are exactly $n$ solutions (roots).
Well, when it comes to quaternions, apparently $i^2=j^2=k^2=-1$, but $i\ne j\ne k\ne i$. So now, we have apparently found three solutions to the second-degree polynomial $x^2=-1$.
I'm not aware of the justification of the FTA, nor I am I aware of Hamilton's justification for quaternions. However, I know a contradiction when I see one. What am I missing here?