$E \times \mathbb C$, where $E$ is a fixed elliptic curve say $\mathbb C^* / \lbrace z \mapsto 2z \rbrace$. It can be compactified as the projective surface $E \times \mathbb P^1$ or as the (non-Kähler) Hopf surface $\mathbb C^2-{0} / \lbrace (z,w) \mapsto (2z,2w) \rbrace$.
Notice that the two compactifications have fields of meromorphic functions of different transcendence degree over $\mathbb C$, i.e., they have different algebraic dimensions.
Another example, different but diffeomorphic to Polizzi's, is given by compactifications of $E \times \mathbb C$, where $E$ is a fixed elliptic curve say $\mathbb C^* / \lbrace z \mapsto 2z \rbrace$. It can be compactified as the projective surface $E \times \mathbb P^1$ or as the (non-Kähler) Hopf surface $\mathbb C^2-{0} / \lbrace (z,w) \mapsto (2z,2w) \rbrace$.
Notice that the two compactifications have fields of meromorphic functions of different transcendence degree over $\mathbb C$, i.e., they have different algebraic dimensions.