The author obtains a generalization of the results of Kronheimer about the existence of hyper-Kähler structures on some adjoint orbits of a complex semisimple Lie group. He proves the following Theorem. (1) Let \(G/H\) be an adjoint orbit of a compact semisimple Lie group \(G\) and \(G^c\), \(H^c\) the complexifications of \(G\), \(H\). Then the manifold \(G^c/H^c\) has a family of complete hyper-Kähler metrics. It is parametrised by triples \((a_1,a_2, a_3)\) consisting of elements from a Cartan subalgebra \(t\) of \(h = \text{Lie }H\) with stabilizer \(H\). The complex manifold \(G^c/H^c\) is isomorphic to the adjoint orbit \(\text{Ad }G^c(a_2 + ia_3)\) if \(H^c\) is the stabilizer of \(a_2 + ai_3\). (2) Let \(b^c \in g^c\) be a nilpotent element which commutes with \(a_2 + ia_3\). Then the complex adjoint orbit \(\text{Ad }G^c(a_2 + ia_3 + b^c)\) has a family of hyper-Kähler metrics parametrised by elements \(a_1 \in t\) such that the centralizers of the pair \((a_2,a_3)\) and the triple \((a_1, a_2, a_3)\) coincide.