  The purpose of this paper is to introduce new parametric families of complex
Hadamard matrices in two different ways. First, we prove that every real
Hadamard matrix of order N>=4 admits an affine orbit. This settles a recent
open problem of Tadej and Zyczkowski, who asked whether a real Hadamard matrix
can be isolated among complex ones. In particular, we apply our construction to
the only (up to equivalence) real Hadamard matrix of order 12 and show that the
arising affine family is different from all previously known examples. Second,
we recall a well-known construction related to real conference matrices, and
show how to introduce an affine parameter in the arising complex Hadamard
matrices. This leads to new parametric families of orders 10 and 14. An
interesting feature of both of our constructions is that the arising families
cannot be obtained via Dita's general method. Our results extend the recent
catalogue of complex Hadamard matrices, and may lead to direct applications in
quantum-information theory.
