Seems like using $\overline{z}$ is cheating if we are not allowed Im. – 2010-12-21
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What is an "expression"? If you allow $\overline{z}$, then you can express Re and Im in terms of it as Rahul writes. If you allow $|z|$ then $|z|^2/z=\overline{z}$ for all nonzero $z$. If you limit yourself to analytic functions, then the answer is "no" as lhf explains. – 2010-12-21