Your question is not clear: what do you mean by "do we need" ? Moreover, inversions are not usually considered as Moebius transformation (they do not preserve orientation), and finally you do not precise what scalars you consider (real or complex).
Yet, I guess that one of the two following facts should answer your question: multiplication by a complex scalar is a similarity (a rotation if the number has modulus $1$), and the composition of two reflexions (which are inversions with respect to lines, which are circles containing the point at infinity) is a rotation.
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