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Let $f:(a+bi) \to (a^2+b)+ai$

Is there an expression for f such that does not use functions $\Re$ and $\Im$?

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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

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No expression as an analytic function because it does not satisfy the Cauchy-Riemann equations.