  We consider the "Mandelbrot set" $M$ for pairs of complex linear maps,
introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and
others. It is defined as the set of parameters $\lambda$ in the unit disk such
that the attractor $A_\lambda$ of the IFS $\{\lambda z-1, \lambda z+1\}$ is
connected. We show that a non-trivial portion of $M$ near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of $M$ are in the closure of the set of interior points of $M$). Next we
turn to the attractors $A_\lambda$ themselves and to natural measures
$\nu_\lambda$ supported on them. These measures are the complex analogs of
much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os
and Garsia, we demonstrate how certain classes of complex algebraic integers
give rise to singular and absolutely continuous measures $\nu_\lambda$. Next we
investigate the Hausdorff dimension and measure of $A_\lambda$, for $\lambda$
in the set $M$, for Lebesgue-a.e. $\lambda$. We also obtain partial results on
the absolute continuity of $\nu_\lambda$ for a.e. $\lambda$ of modulus greater
than $\sqrt{1/2}$.
