  We provide a self-similar measure for the self-similar group $G$ acting
faithfully on the binary rooted tree, defined as the iterated monodromy group
of the quadratic polynomial $z^2+i$. We also provide an $L$-presentation for
$G$ and calculations related to the spectrum of the Markov operator on the
Schreier graph of the action of $G$ on the orbit of a point on the boundary of
the binary rooted tree.
