We consider complex Hénon maps that have a semi-indifferent fixed point with eigenvalues $w_1$ and $w_2$, where $|w_1|=1$ and $|w_2|<1$. At a semi-parabolic parameter (i.e. when $w_1$ is a root of unity) we have a good understanding of this family: for small Jacobian, the dynamics of the Julia set of the Hénon map fibers over the dynamics of a polynomial Julia set. This is joint work with R. Tanase. The situation when $w_1=\exp{(2\pi i t)}$ and $t$ is irrational is more complex as it depends on the arithmetic properties of $t$. When $t$ is the golden mean, we show that the Hénon map with small Jacobian has a Siegel disk whose boundary is homeomorphic to a circle. The proof is based on renormalization of commuting pairs. This is joint work with D. Gaydashev and M. Yampolsky. We will also explain where these maps sit in the whole parameter space of complex Hénon maps and explore other directions.