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What would the final Quaternion Solutions look like for $x^2+1=0$?

I substituted in $x = a+bi+cj+dk$ and came up with a very long +/- square root.

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The solutions have the form $$ bi + cj + dk $$ where $b^2 + c^2 + d^2 = 1$. They are the "pure imaginary unit quaternions".

Why? Because $x^2 = -1$, so $|x|^2 = 1$ so $a^2 + b^2 + c^2 + d^2 = 1$. Since the real part of $x^2$ is $a^2 - b^2 - c^2 - d^2 = -1$, we get that $2a^2 = 0$, so $a = 0$.

Once you settle that, it's easy to check that all the other pure-imaginary unit quaternions do in fact square to $-1$.

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