Quaternion roots for $x^2+2x+2$

Question

So I know there are complex solutions $-1+i$ and $-1-i$ I'm not entirely sure how to describe all quaternion solutions. I know $x^2=-1$ has infinitely many solutions in the quaternions.

I also know you can rotate quaternions, if I take a solution and rotate it around some vector, do those rotations also solve the equation?

Answer 1

Keep calm and complete the square. To wit:

$x^2+2x=-2$

$x^2+2x+1=-2+1$

$(x+1)^2=-1$

So $x+1$ is an imaginary unit $u$, described in the usual quaternion notation as $ai+bj+ck$ with $a,b,c$ any real numbers satisfying $a^2+b^2+c^2=1$. Then, $x=-1+u$.