  We derive the Hamiltonian tangential Carleman approximation for the complex
Calogero-Moser space $\mathcal{C}_n$ with a non-compact real form
$\mathcal{C}_n^{\mathbb{R}}$, using the symplectic density property of
$\mathcal{C}_n$. The Hamiltonian tangential Carleman approximation means to
approximate symplectic diffeomorphisms of $\mathcal{C}_n^{\mathbb{R}}$ which
are smoothly isotopic to the identity by symplectic holomorphic automorphisms
of $\mathcal{C}_n$ which in addition preserve the real form
$\mathcal{C}_n^{\mathbb{R}}$. This appproximation is in the strongest topology,
in the fine Whitney topology.
