The first thing to realize is that the question must be properly interpreted, which probably can be done in several ways.

It seems reasonable to interpret it as follows:

Find a ring $R$ with an injective homomorphism $\mathbb{C} \rightarrow R$, such that:

• there is an involution $\sigma: R \rightarrow R$, i.e. an endomorphism of order $2$, such that the restriction of $\sigma$ to $\mathbb{C}$ is the complex conjugation;

• there is an element $f \in R$, such that $f \sigma(f) = -1$;
• $R$ is generated by $\mathbb{C}$ and $f$.

But you probably also want to put other conditions on $R$.

For example, if you require $R$ to be commutative, then $R$ can be the ring $\mathbb{C} \oplus \mathbb{C}$, with $\mathbb{C}$ embedded diagonally, $\sigma(z, w)=(\overline{w}, \overline{z})$, and the element $f=(1, -1)$.

The first thing to realize is that the question must be properly interpreted, which probably can be done in several ways.

It seems reasonable to interpret it as follows:

Find a ring $R$ with an injective homomorphism $\mathbb{C} \rightarrow R$, such that:

• there is an involution $\sigma: R \rightarrow R$, i.e. an endomorphism of order $2$, such that the restriction of $\sigma$ to $\mathbb{C}$ is the complex conjugation;

• there is an element $f \in R$, such that $f \sigma(f) = -1$;
• $R$ is generated by $\mathbb{C}$ and $f$.

The last condition can possibly be replace by: $R$ is generated by $\mathbb{C}$, $f$ and $\sigma(f)$.

But you probably also want to put other conditions on $R$.

For example, if you require $R$ to be commutative, then $R$ can be the ring $\mathbb{C} \oplus \mathbb{C}$, with $\mathbb{C}$ embedded diagonally, $\sigma(z, w)=(\overline{w}, \overline{z})$, and the element $f=(1, -1)$.

If you don't require $R$ to be commutative, then it can also be the biquaternion $R = \mathbb{C} \oplus \mathbb{C} i \oplus \mathbb{C} j \oplus \mathbb{C} k$.

For the moment I don't have an example where $R$ is still a division algebra and $f$ is algebraic.

The first thing to realize is that the question must be properly interpreted, which probably can be done in several ways.

It seems reasonable to interpret it as follows:

Find a ring $R$ with an injective homomorphism $\mathbb{C} \rightarrow R$, such that:

• there is an involution $\sigma: R \rightarrow R$, i.e. an endomorphism of order $2$, such that the restriction of $\sigma$ to $\mathbb{C}$ is the complex conjugation;

• there is an element $f \in R$, such that $f \sigma(f) = -1$;
• $R$ is generated by $\mathbb{C}$ and $f$.

But you probably also want to put other conditions on $R$.

For example, if you require $R$ to be commutative, then $R$ can be the ring $\mathbb{C} \oplus \mathbb{C}$, with $\mathbb{C}$ embedded diagonally, $\sigma(z, w)=(\overline{w}, \overline{z})$, and the element $f=(1, -1)$.

Or maybe you want to keep $R$ a division algebra (i.e. all non-zero elements are invertible). Then it can be the quaternion algebra $\mathbb{H} = \mathbb{C} \oplus \mathbb{C}j$, with the usual embedding $\mathbb{C}\rightarrow\mathbb{H}$, the usual involution $\sigma(z + wj) = \overline{z} - j\overline{w}$, and the element $f = j$.

The first thing to realize is that the question must be properly interpreted, which probably can be done in several ways.

It seems reasonable to interpret it as follows:

Find a ring $R$ with an injective homomorphism $\mathbb{C} \rightarrow R$, such that:

• there is an involution $\sigma: R \rightarrow R$, i.e. an endomorphism of order $2$, such that the restriction of $\sigma$ to $\mathbb{C}$ is the complex conjugation;

• there is an element $f \in R$, such that $f \sigma(f) = -1$;
• $R$ is generated by $\mathbb{C}$ and $f$.

But you probably also want to put other conditions on $R$.

For example, if you require $R$ to be commutative, then $R$ can be the ring $\mathbb{C} \oplus \mathbb{C}$, with $\mathbb{C}$ embedded diagonally, $\sigma(z, w)=(\overline{w}, \overline{z})$, and the element $f=(1, -1)$.