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@MvG, I would assume that there might be more freedom than just "you can map any 3 points to any 3 points," although it is certainly not true that you can map any 4 points to any other 4 points (which is really what you would hope to be true), so it would be good to figure out precisely what degrees of freedom exist. Also, there is something odd here, because the action of $PSL_2(\mathbb{R})$ is determined by three points and so is $PSL_2(\mathbb{C})$. Algebraically, it feels like $PSL_2(\mathbb{H})$ should behave similarly---granted, we probably lose a lot jumping to the non-commutative case.
