  Motivated by the construction based on topological suspension of a family of
compact non-K\"ahler complex manifolds with trivial canonical bundle given by
L. Qin and B. Wang in [QW], we study toric suspensions of balanced manifolds by
holomorphic automorphisms. In particular, we show that toric suspensions of
Calabi-Yau manifolds are balanced. We also prove that suspensions associated
with hyperbolic automorphisms of hyperk\"ahler manifolds do not admit any
pluriclosed, astheno-K\"ahler or p-pluriclosed Hermitian metric. Moreover, we
consider natural extensions for hypercomplex manifolds, providing some explicit
examples of compact holomorphic symplectic and hypercomplex non-K\"ahler
manifolds. We also show that a modified suspension construction provides
examples with pluriclosed metrics.
