  Rational minimax approximation of real functions on real intervals is an
established topic, but when it comes to complex functions or domains, there
appear to be no algorithms currently in use. Such a method is introduced here,
the {\em AAA-Lawson algorithm,} available in Chebfun. The new algorithm solves
a wide range of problems on arbitrary domains in a fraction of a second of
laptop time by a procedure consisting of two steps. First, the standard AAA
algorithm is run to obtain a near-best approximation and a set of support
points for a barycentric representation of the rational approximant. Then a
"Lawson phase" of iteratively reweighted least-squares adjustment of the
barycentric coefficients is carried out to improve the approximation to
minimax.
