I know that the algebra of quaternions is isomorphic to the division algebra of 2×2 complex matrices which are real multiples of the elements of SU(2). I also know that SU(2) is a double cover of SO(3).
The scalar multiples of the elements of SO(3) also form a group (or even an algebra?). Is there a name for this group (or algebra)?
The clearest name is just "rotation-dilation group". Using, as suggested, the name "conformal group" will only work if explicitly restricted to linear conformal transformations, otherwise one might immediately think of the Möbius group, which does include rotations and dilations but also other transformations.