Is there such an operation? What kind of number would the answer be?
There is an operation between real numbers whose answer is not real: $\sqrt{-1} = (-1)^{1/2} = i$. However, operations between complex numbers such as $\sqrt{i} = i^{1/2} = e^{i\pi/4}$ and $i^i = e^{-\pi/2}$ (principal value) result in complex numbers.
It is known that the solutions of algebraic equations with complex coefficients are always complex numbers.
The values of functions such as $i! = \Gamma(1 + i) \approx 0.4980 - 0.1549i$ are also complex numbers. Is there any function whose value is not complex for a complex input?
One thing that came to my mind is $^i i = \underbrace {i^{i^{\cdot^{\cdot^i}}}}_{i \text{ times}}$, the "$i$th" tetration of $i$, but according to this one, although I don't understand it, the value still seems to be complex.
Edit:
I found a duplicate question. The answer there seems to be "there is no such algebraic operation by the fundamental theorem of algebra."
My question is not limited to algebraic operations, but I can't seem to find any such mathematical function. Then the question arises as to why there are none.
I wonder if this is because when we extend numbers from complex numbers to, say, quaternions, we lose properties such as commutativity, so mapping to such a larger domain is not very useful.