The corresponding question for the real special orthogonal group is well-known - these are the Spin groups. When one looks at complex special orthogonal groups though, this isn't the right way to go: Fundamental Group of Spin^c(V) are non-trivial, and in fact contain a copy of $\mathbb{Z}$.
For low-ish numbers, we can say something.
When $n=1$, $SO(1,\mathbb{C})$ is trivial.
Next, for $n=2$, this is just $\mathbb{C}^\ast$, so the fundamental group is the circle group and the universal cover is the Riemann surface corresponding to $\log$, with its natural group structure.
For $n=3$ we can think of $\mathbb{C}^3$ as the space of traceless $2\times 2$ matrices. In this setting, $g \in SL(2, \mathbb{C})$ acts by conjugation, and this gives a double cover of $SO(3, \mathbb{C})$, which reveals the universal cover, as $SL(2,\mathbb{C})$ is simply connected. It is in fact realizable as $S^3 \times H^3$, which uses the polar decomposition to realize the elements of $SL(2, \mathbb{C})$ as unitary matrices together with positive semi-definite, self-adjoint, special linear matrices. This can be used to obtain the equation of the hyperboloid, so that one obtains the product space above.
I have no idea what happens for $n > 3$ though. Is a general description possible? I know that $\pi_1(SO(n, \mathbb{C})) \simeq \mathbb{Z}/2\mathbb{Z}$ for all $n> 2$, as it says so on Wikipedia, but it evades me for a moment on how to prove this. I'll think more about that though since it's probably similar to the real case.
Update: I did figure out why that's true, with the help of a friend. $SO(n,\mathbb{C})$ is the complexification of $SO(n,\mathbb{R})$, and so the latter is a maximal compact subgroup of the former. Thus we get a deformation retraction and since $n>2$, this preserves the fundamental group.
Maybe I should clarify the question. I know that there is going to be an exact sequence coming from covering space theory:
$$1 \to \mathbb{Z}/2\mathbb{Z} \to \widetilde{SO(n,\mathbb{C})} \to SO(n, \mathbb{C}) \to 1 $$
What would be cool is maybe something that explains this in terms of Clifford algebras? I know what they are, but not much about them. Can we describe this object analogously to the way Spin is described there?
For $n \ge 3$ the universal cover is a group called the complex spin group $\text{Spin}(n, \mathbb{C})$. This is not the same group as $\text{Spin}^c(n)$, which is confusingly also sometimes called the "complex spin group," and which as you say is not simply connected. We have low-dimensional exceptional isomorphisms
$$\text{Spin}(3, \mathbb{C}) \cong SL_2(\mathbb{C})$$ $$\text{Spin}(4, \mathbb{C}) \cong SL_2(\mathbb{C}) \times SL_2(\mathbb{C})$$ $$\text{Spin}(5, \mathbb{C}) \cong \text{Sp}(4, \mathbb{C})$$ $$\text{Spin}(6, \mathbb{C}) \cong SL_4(\mathbb{C})$$
which complexify corresponding exceptional isomorphisms for the real Spin groups; see, for example, these notes by Seewoo Lee.