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Since it doesn't really involve the complex structure, can't we use the standard real coordinates on $\mathbb{C}^2=\mathbb{R}^4$ as $x_0,x_1,x_2,x_3$ and consider the harmonic polynomial $$ f = x_0 + i\ (3{x_0}^2-{x_1}^2-{x_2}^2-{x_3}^2)? $$ This doesn't vanish on the unit sphere and doesn't seem (even under linear changes of variables) to be of the kind you seem to want to avoid (though you didn't define it very carefully). If you want to re-express it in terms of $z, \bar z, w,\bar w$ where $z = x_0+ix_1$ and $w = x_2 + ix_3$, you can certainly do that.