We introduce the concepts of a strong implicative hyper $BCK$-ideal and a strong positive implicative hyper $BCK$-ideal in hyper $BCK$-algebras, and investigates some related properties. Also we introduce the notions of a maximal hyper $BCK$-ideal and boundedness in hyper $BCK$-algebras, and investigates some related properties. First, we show that every strong (positive) implicative hyper $BCK$-ideal is a strong hyper $BCK$-ideal and (positive) implicative hyper $BCK$-ideal, but the converse is not true. Also we obtain the equivalent condition of strong implicative hyper $BCK$-ideals and strong hyper $BCK$-ideals, and we discuss the relations between strong positive implicative hyper $BCK$-ideals and strong implicative hyper $BCK$-ideals. Next, we show that if $H$ is a bounded hyper $BCK$-algebra and $|H|\geq 2$, then $H$ has at least one maximal strong hyper $BCK$-ideal, and if $H$ is a bounded hyper $BCK$-algebra and $I$ is a proper strong (positive) implicative hyper $BCK$-ideal of $H$, then there is a maximal strong (positive) implicative hyper $BCK$-ideal containing $I$.