The authors construct the complex powers \(A_{\alpha}\) (\(\alpha\in \mathbb{C}_+\)) of some classes of closed linear operators such as almost \(C\)-nonnegative operators and \((a,b,C)\)-nonnegative operators with \(a,b\in\mathbb{R}\) in the framework of sequentially complete locally convex spaces. Some applications of the obtained results to abstract incomplete problems involving the generators of integrated or \(C\)-regularized semigroups are also presented.