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Browsing this question: Why are the solutions of polynomial equations so unconstrained over the quaternions?, the pdf linked in the comments says that the infinitely many conjugates of $i$ in $\mathbb{H}$ are roots to $x^2+1$.

I get that they're roots, but how do we know that the conjugacy class of $i$ is in fact infinite?

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    Have you tried writing down the conjugacy class of $i$?2012-07-11
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    Calculate $$(\cos\alpha +\sin\alpha j)^{-1}i(\cos\alpha +\sin\alpha j).$$2012-07-11

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