Our aim in this paper is to prove that every separable infinite dimensional complex Banach space admits a topologically mixing holomorphic uniformly continuous semigroup and to characterize the mixing property for semigroups of operators. A concrete characterization of being topologically mixing for the translation semigroup on weighted spaces of functions is also given. Moreover, we prove that there exists a commutative algebra of operators containing both, a chaotic operator and an operator $T$, so that $T$ is not a multiple of the identity and no multiple of $T$ is chaotic. This gives a negative answer to a question of deLaubenfels and Emamirad.