With $\langle x,y\rangle=x^*y$ and $a\in\mathbb C$, $\langle ax,y\rangle=\overline a\langle x,y\rangle$, while $\langle x,ay\rangle=a\langle x,y\rangle$. Thus the inner product is sesquilinear, not complex-bilinear. For $x=y=1$ and $a=i$, the two results are $-i$ and $i$.
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