Think about the easier question of classifying one-dimensional complex Lie algebras: there's only one! -- namely, $\mathbb{C}$ with the trivial bracket. The simply connected Lie group with $\mathbb{C}$ as its Lie algebra is $\mathbb{C}$ itself, and thus all other *connected* complex Lie groups with Lie algebra $\mathbb{C}$ are quotients of $\mathbb{C}$ by a discrete subgroup $\Gamma \subset \mathbb{C}$. This subgroup $\Gamma$ is a lattice in $\mathbb{C}$ and is therefore classified by rank: if $\mathrm{rank} \, \Gamma = 1$, then $\mathbb{C}/\Gamma = \mathbb{C}^\ast$; and if $\mathrm{rank} \, \Gamma = 2$, then $\mathbb{C}/\Gamma$ is a Riemann surface of genus one (and by varying $\Gamma$ all such arise).

couldhave been asking for just the topological types of connected complex Lie groups of dimension 1 over the complex numbers (btw, $\mathbb{C}^\ast$ is a cylinder), but it also sounds as if he might not have been aware that the topology is too coarse to distinguish them complex-analytically. – Todd Trimble♦ Nov 28 '10 at 12:52