I'm not sure if it this is right: Imagen that I have this quaternion $a + bi + cj + dk$ where $a, b, c, d \in \mathbb{R}$.
Then I take $cj + dk$ and make this $(cj + dk)^2=-c^2 +cdi-dci-d^2=-c^2-d^2$ so $(cj + dk)=\pm\sqrt{-c^2-d^2}=\pm i\sqrt{c^2+d^2}$
Then $a + bi + cj + dk = a + bi \pm i\sqrt{c^2 + d^2} = a + i(b \pm \sqrt{c^2+d^2})$
So a Quaternion is just an ordinary complex number?? $a + i(b + \sqrt{c^2+d^2})$ or $a + i(b - \sqrt{c^2+d^2})$ , I mean, it is only an ordinary number $x+yi$
Am I doing something wrong here?
Thank you
What you can do is render $cj+dk$ as $u\sqrt{c^2+d^2}$ where $u$ is an imaginary unit, that is $u$ is a quaternion solution to $u^2=-1$. But, as @lulu points out, quaternions have more than one additive inverse pair of such units, and in this case $i$ and $-i$ are just not the right pair.