I know that every möbius transform that preserves the upper half plane is of the form
$m(z) = \frac{az+b}{cz+d}$, where $a,b,c,d \in \mathbb{R}$, or $m(z) = \frac{a\bar{z} + b}{c\bar{z} + d}$, where $a,b,c,d$ are all purely imaginary. In both cases $ad-bc = 1$.
However, I could be wrong but I see that most books in the literature tend to exclude the latter case involving the complex conjugate. Is there any reason why??
It is pretty important that I understand this, as I need to use these facts when calculating the distance between two points in the upper half plane model of 2 dimensional hyperbolic space.