  We study higher cluster tilting objects in generalized higher cluster
categories arising from dg algebras of higher Calabi-Yau dimension. Taking
advantage of silting mutations of Aihara-Iyama, we obtain a class of
$m$-cluster tilting objects in generalized $m$-cluster categories. For
generalized $m$-cluster categories arising from strongly ($m+2$)-Calabi-Yau dg
algebras, by using truncations of minimal cofibrant resolutions of simple
modules, we prove that each almost complete $m$-cluster tilting $P$-object has
exactly $m+1$ complements with periodicity property. This leads us to the
conjecture that each liftable almost complete $m$-cluster tilting object has
exactly $m+1$ complements in generalized $m$-cluster categories arising from
$m$-rigid good completed deformed preprojective dg algebras.
