Under certain hypotheses, diffusion semigroups on commutative $L^p$-spaces are known to have an analytic extension for $|\arg z| < \pi/2 −\arctan (|p−2| / 2 \sqrt{p−1})$. In this paper it is shown that semigroups on non-commutative $L^p$-spaces have the same extension under suitable conditions. These conditions even lead to a new result in the commutative case. Further, some examples are considered.