By considering fixed points in $\mathbb{C} \cup\{\infty\}$, prove that any complex Möbius transformation is conjugate either to a map of the form $z \mapsto k z$ for some $k \in \mathbb{C}$ or to $z \mapsto z+1$. Deduce that two Möbius transformations $g, h$ (neither the identity) are conjugate if and only if $\operatorname{tr}^{2}(g)=\operatorname{tr}^{2}(h)$.

Does every Möbius transformation $g$ also have a fixed point in $\mathbb{H}^{3}$ ? Briefly justify your answer.