Solution
= Solution
Write $\alpha=a+ib$ with $b>0$. Then
$$
\boxed{f_\alpha(z)=a+bz.}
$$
It preserves the <upper half-plane>, has inverse $(z-a)/b$, and sends $i$ to $\alpha$. Its nonzero <derivative> $b$ verifies that this <bijection> is conformal.