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.
The motivation is described in the accepted answer to the question that you mentioned. On the other hand, the field $\mathbb C$ of complex numbers is algebraically closed. This means that any nonconstant polynomial function from $\mathbb C$ into itself has a complex root. Actually, it can be written as the product of first degree polynomials. So, the need to create the quaternions was not due to something lacking in $\mathbb C$, at least from the algebraic point of view.