Is there an operation in complex numbers that can only be answered by quaternions?

The natural numbers cannot provide an answer to $1-2$.

The integers cannot provide an answer to $\frac{1}{2}$.

The rational numbers cannot provide an answer to $\sqrt{2}$.

The real numbers cannot provide an answer to $\sqrt{-1}$.

The complex numbers cannot provide an answer to what (leading to quaternions)? Is this the way it works?

This question asks similar. The accepted answer points to the Cayley-Dickson construction but that doesn't seem to address an operation between complex numbers that cannot be a complex number.