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.

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.

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.

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.