I'm having trouble coming up with an easy-to-describe surjection $f : \mathbb{R} \to \mathbb{C}$. Here's what I came up with: (Edit: this doesn't work; see Qiaochu's comment)
Define $P(x \in \mathbb{R}) = \frac{1}{1-e^{-x}}$ and $P^{-1}(x \in (0,1)) = -\log \frac{1-x}{x}$.
Let $f(x) = P(\operatorname{even}(P^{-1}(x))) \cdot e^{2 \pi i \operatorname{odd}(P^{-1}(x))}$, where
$$\operatorname{even}(0.b_0b_1b_2\ldots) = 0.b_0b_2b_4\ldots, \\\operatorname{odd}(0.b_0b_1b_2\ldots) = 0.b_1b_3b_5\ldots.$$
However, this seems needlessly complicated. Is there a simpler way?