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I just want to know an example of non-constant complex valued function which is open map.

Is this ok?

$f:\mathbb{C}\rightarrow \mathbb{C}$ given by $f(z)=\bar{z}$? This is just a reflection with respect to $x$-axis, right?

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    Why not take any analytic non-constant function?2012-06-29

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Compiling the comments: $f(z)=\bar z$, or $f(z)=z$, or any nonconstant holomorphic function will work.