  We discuss the possibility of extending the RG flows to complex coupling
spaces. We argue that the Fisher's zeros are located at the boundary of the
complex basin of attraction of IR fixed points. We support this picture with
numerical calculations at finite volume for2D O(N) models in the large-N limit
and the hierarchical Ising model using the two-lattice matching method. We
present numerical evidence supporting the idea that, as the volume increases,
the Fisher's zeros of 4-dimensional pure gauge SU(2) lattice gauge theory with
a Wilson action, stabilize at a distance larger than 0.1 from the real axis in
the complex beta=4/g^2 plane. We show that when a positive adjoint term is
added, the zeros get closer to the real axis. We compare the situation with the
U(1) case. We discuss the implications of this new framework for proofs of
confinement and searches for nontrivial IR fixed points in models beyond the
standard model.
