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I'm trying to find an example of a function that is harmonic but not pluriharmonic. It's trivial to show that pluriharmonic implies harmonic, but I want to show that the reverse direction is not true.

For reference, we define a pluriharmonic function, to be a function $f$ such that $$\frac{\partial^2 f}{\partial z_{\nu} \partial \overline{z}_{\nu}} = 0, \ \ \ \nu = 1,..., n.$$

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    You want the derivative for $z_\nu$ and $\bar{z}_\mu$ independently. For example, $z_1\bar{z}_2$ is not pluriharmonic, but it satisfies your definition. – 2017-01-31

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For $(z,w)\in\mathbb C^2,$ let's write $z=u+iv,w=x+iy.$ Let $f(z,w) = u^2 + v^2 - 2x^2.$ Then $f$ is harmonic on $\mathbb C^2 = \mathbb R^4.$ But $z\to f(z,0) = u^2 + v^2$ is not harmonic as a function on $\mathbb C = \mathbb R^2.$