Gaussian rational field

Here $i^2=-1$, and conjugation sends $a+bi$ to $a-bi$. The subring with integral coordinates is the Gaussian integer ring.

A rational-linear endomorphism is any two-by-two rational matrix in the basis $1,\alpha$, where $\alpha$ is the displayed quadratic generator. A unit-preserving rational-algebra map must send $\alpha$ to a root of its defining polynomial, so there are precisely the identity and conjugation. The invertible linear maps form $\mathrm{GL}_2(\mathbb Q)$, a much larger group.