For $x,y\in\mathbb C^n$, use $\langle x,y\rangle=\sum_j\overline{x_j}y_j=x^*y$. It is conjugate-linear in $x$, linear in $y$, and satisfies $\langle y,x\rangle=\overline{\langle x,y\rangle}$. Example: $\langle(1,i),(1,i)\rangle=2$, whereas the unconjugated product would be zero.
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