In this paper we study some properties of the adjusted normal cone operator of quasiconvex functions. In particular, we introduce a new notion of maximal quasimotonicity for set-valued maps different from similar ones recently appeared in the literature, and we show that it is enjoyed by this operator. Moreover, we prove the $s\times w^*$ cone upper semicontinuity of the normal cone operator in the domain of $f$ in case the set of global minima has non empty interior.