= Solution
By point transitivity, it suffices to take $P=i$. The condition
$$
\frac{ai+b}{ci+d}=i
$$
is equivalent to $d=a$, $c=-b$, and the determinant condition becomes $a^2+b^2=1$. Thus the stabilizer consists of
$$
\begin{pmatrix}a&b\\-b&a\end{pmatrix},
\qquad a^2+b^2=1,
$$
modulo $\pm I$. Writing $a=\cos\varphi$, $b=\sin\varphi$ identifies it with rotations of the tangent plane, and the doubled matrix angle removes the $\pm I$ quotient. Hence
$$
\boxed{\operatorname{Stab}_G(P)\cong SO(2)}.
$$
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