  We consider some new limits for the elliptic hypergeometric integrals on root
systems. After the degeneration of elliptic beta integrals of type I and type
II for root systems $A_n$ and $C_n$ to the hyperbolic hypergeometric integrals,
we apply the limit $\omega_1\to - \omega_2$ for their quasiperiods
(corresponding to $b\to i$ in the two-dimensional conformal field theory) and
obtain complex beta integrals in the Mellin--Barnes representation admitting
exact evaluation. Considering type I elliptic hypergeometric integrals of a
higher order obeying nontrivial symmetry transformations, we derive their
descendants to the level of complex hypergeometric functions and prove the
Derkachov--Manashov conjectures for functions emerging in the theory of
non-compact spin chains. We describe also symmetry transformations for a type
II complex hypergeometric function on the $C_n$-root system related to the
recently derived generalized complex Selberg integral. For some hyperbolic beta
integrals we consider a special limit $\omega_1\to \omega_2$ (or $b\to 1$) and
obtain new hypergeometric identities for sums of integrals of rational
functions.
