  We provide a general construction procedure for antilinearly invariant
complex root spaces. The proposed method is generic and may be applied to any
Weyl group allowing to take any element of the group as a starting point for
the construction. Worked out examples for several specific Weyl groups are
presented, focusing especially on those cases for which no solutions were found
previously. When applied in the defining relations of models based on root
systems this usually leads to non-Hermitian models, which are nonetheless
physically viable in a self-consistent sense as they are antilinearly invariant
by construction. We discuss new types of Calogero models based on these complex
roots. In addition we propose an alternative construction leading to q-deformed
roots. We employ the latter type of roots to formulate a new version of affine
Toda field theories based on non-simply laced roots systems. These models
exhibit on the classical level a strong-weak duality in the coupling constant
equivalent to a Lie algebraic duality, which is known for the quantum version
of the undeformed case.
