Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?

Question

Excluding polynomials (whose solutions are covered by the Fundamental Theorem of Algebra), do there exist any univariable equations that cannot be solved in the complex numbers, but can be solved other fields, such as the quaternions, octonions, etc.?

I know that not all equations can be solved (in any field), such as $e^w=0,$ or $0x=0$ (no unique solution), and also that some equations that can be solved in $\mathbb{C}$ can also be solved in $\mathbb{H}$, but with many more solutions (e.g. $w^2+1=0$), but I'm wondering whether there exist equations that can only be solved in higher-dimensional division algebras (than $\mathbb{C}$)?

Thanks

Answer 1

$(\textrm{i} \, x - x \, \textrm{i})^2+1=0$