Fun examples relating to Hopf surfaces A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following examples:
(i) A non-compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. 
(ii) A compact complex surface whose universal cover is smoothly isomorphic to $\mathbb{C}^2 \setminus \{ 0  \}$, but not complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. 
 A: (ii) I would like to prove that there are no complex surfaces that satisfy (ii).
Indeed, suppose that the universal cover $\widetilde X$ of a complex surface $X$ is diffemorphic to $\mathbb C^2\setminus 0$. Let's prove that $\widetilde X$ is biholomorphic to $\mathbb C^2\setminus 0$.
First, we note that $X$ has a finite cover that is diffeomorphic to $S^3\times S^1$. Indeed, take any 
$$S^3\subset \widetilde X=\mathbb C^2\setminus 0$$ 
that encircles $0$. Then, since the action of $\pi_1(X)$ on $\mathbb C^2\setminus X$ is discreet, there exists only finite number of elements $g\in \pi_1(X)$ such that $g\cdot S^3$ intersects $S^3$ in $\mathbb C^2\setminus 0$. Let's take such $g_1\in\pi_1(X)$ that $g_1\cdot S^3$ is disjoint from $S^3$. Let $\mathbb Z=\langle g_1\rangle $ be the group generated by $g_1$. Then it is not hard to see that $(\mathbb C^2\setminus 0)/\langle g_1\rangle$ is diffeomorphic to $S^1\times S^3$. Clearly  $S^1\times S^3$ is a finite cover of $X$.
It remains now to apply the result of Bogomolov that complex surfaces with $b_2=0$ are either Hopf surfaces or Inoue surfaces https://en.wikipedia.org/wiki/Surface_of_class_VII . Since $\pi_1(S^1\times S^3)=\mathbb Z$, the complex structure on $S^1\times S^3$ is that of a Hopf surface. We conclude that $\widetilde X$ is byholomorphic to $\mathbb C^2\setminus 0$.
(i) As for (i), you can take $(\mathbb C^2\setminus 0)/\Gamma$, where $\Gamma$ is any finite subgroup of $SU(2)$.
