By one of those coincidences so common in mathematics, two days after I asked this question, something unrelated I was thinking about led me to this wonderful paper of Hanspeter Kraft, which points out (among many other things) that $\mathbb{C}^3$ has other algebraic structures, for example the hypersurface in $\mathbb{A}^4$ given by $x + x^2 y + z^3 + t^2 = 0$ (Peter Russell showed it was analytically $\mathbb{C}^3$, and Makar and Limanov Makar-Limanov showed that it is not algebraically isomorphic to $\mathbb{A}^3$).
By one of those coincidences so common in mathematics, two days after I asked this question, something unrelated I was thinking about led me to this wonderful paper of Hanspeter Kraft, which points out (among many other things) that $\mathbb{C}^3$ has other algebraic structures, for example the hypersurface in $\mathbb{A}^4$ given by $x + x^2 y + z^3 + t^2 = 0$ (Peter Russell showed it was analytically $\mathbb{C}^3$, and Makar and Limanov showed that it is not algebraically isomorphic to $\mathbb{A}^3$).