= Solution
With conjugation in its first slot, a complex <inner product> has <conjugate symmetry>, <linearity> in its second slot, and <positive definiteness>. For the proposed <Hermitian inner product>, direct expansion gives
$$
\langle z,y\rangle=\overline{\langle y,z\rangle},\qquad
\langle z,\alpha y+\beta w\rangle=\alpha\langle z,y\rangle+\beta\langle z,w\rangle,\qquad
\langle z,z\rangle=\sum_j|z_j|^2\ge0.
$$
The last expression vanishes exactly for the <zero vector>. <Conjugate symmetry> and <linearity> also give <antilinearity> in the first slot: $\langle\alpha z+\beta w,y\rangle=\bar\alpha\langle z,y\rangle+\bar\beta\langle w,y\rangle$. These verify all three axioms and the associated first-slot consequence.
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