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A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if $f$ is a holomorphic function of $\mathbb{C}^{n}$ that vanishes in a real hyperplane, then $f$ is identically zero?

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The zero locus of an analytic function of real codimension 2 unless the function is identically zero. I suppose you may need the Weierstrass preparation theorem in order to give a rigorous proof. –  Andrew Oct 16 '12 at 14:50
You might consider the following. Let $L$ be a one (complex) dimensional subspace of $\mathbb{C}^n$. You know that $f|_L$ is holomorphic on $L\cong \mathbb{C}$. What does the zero set of $f$ look like in $L$? It certainly contains the intersection of $L$ and Re$(\ell(Z)) = c$. How big is this set? –  froggie Oct 16 '12 at 15:23

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