Uniqueness of the Complex Numbers: Every Field Generated over an Embedded Copy of the Reals by a Square Root of -1 Is Uniquely Isomorphic to C

In the setting of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers, let $K$, with $+$, $\cdot$, $0_K$ and $1_K$, be a field; let $\lambda:\mathbb{R}\to K$ be an injective map with $\lambda(a+b)=\lambda(a)+\lambda(b)$ and $\lambda(ab)=\lambda(a)\cdot\lambda(b)$ for all $a,b\in\mathbb{R}$ and $\lambda(1)=1_K$; and let $u\in K$ satisfy $u\cdot u=-1_K$ and be such that every element of $K$ equals $\lambda(a)+\lambda(b)\cdot u$ for some $a,b\in\mathbb{R}$.

¶ There is exactly one map $\varphi:\mathbb{C}\to K$ with $\varphi(z+w)=\varphi(z)+\varphi(w)$ and $\varphi(zw)=\varphi(z)\cdot\varphi(w)$ for all $z,w\in\mathbb{C}$, $\varphi(a)=\lambda(a)$ for every $a\in\mathbb{R}$, and $\varphi(i)=u$.

¶ This $\varphi$ is a bijection, and its inverse $\varphi^{-1}:K\to\mathbb{C}$ satisfies $\varphi^{-1}(x+y)=\varphi^{-1}(x)+\varphi^{-1}(y)$ and $\varphi^{-1}(x\cdot y)=\varphi^{-1}(x)\varphi^{-1}(y)$ for all $x,y\in K$, $\varphi^{-1}(\lambda(a))=a$ for every $a\in\mathbb{R}$, and $\varphi^{-1}(u)=i$.

¶ Let $K'$, with $+$, $\cdot$, $0_{K'}$ and $1_{K'}$, be a field, let $\lambda':\mathbb{R}\to K'$ be a map and let $u'\in K'$, and suppose that $K'$, $\lambda'$ and $u'$ satisfy the hypotheses above with $K'$, $\lambda'$, $u'$, $0_{K'}$ and $1_{K'}$ in place of $K$, $\lambda$, $u$, $0_K$ and $1_K$. Then there is exactly one map $\psi:K\to K'$ with $\psi(x+y)=\psi(x)+\psi(y)$ and $\psi(x\cdot y)=\psi(x)\cdot\psi(y)$ for all $x,y\in K$, $\psi(\lambda(a))=\lambda'(a)$ for every $a\in\mathbb{R}$, and $\psi(u)=u'$. This $\psi$ is a bijection, and its inverse $\psi^{-1}:K'\to K$ has the corresponding properties with the roles exchanged: $\psi^{-1}(x'+y')=\psi^{-1}(x')+\psi^{-1}(y')$ and $\psi^{-1}(x'\cdot y')=\psi^{-1}(x')\cdot\psi^{-1}(y')$ for all $x',y'\in K'$, $\psi^{-1}(\lambda'(a))=\lambda(a)$ for every $a\in\mathbb{R}$, and $\psi^{-1}(u')=u$.