Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb{C}$.
If $\|x\|=\sqrt{\langle x,x\rangle}$ and $\|x\|=\sqrt{\langle x,x\rangle'}$ for all $x\in X$ where $\langle,\rangle$ and $\langle,\rangle'$ are complex inner products, then must we have $\langle,\rangle=\langle,\rangle'$ ?
If $X$ is a vector space over $\mathbb{R}$, I think the answer is yes. Indeed, since $$ \forall x\in X:\|x\|=\sqrt{\langle x,x\rangle}=\sqrt{\langle x,x\rangle'}\implies\forall x\in X:\langle x,x\rangle=\langle x,x\rangle' $$ then for $x,y\in X$ we have $\langle x+y,x+y\rangle=\langle x+y,x+y\rangle'$ and expanding we see that $\langle x,y\rangle=\langle x,y\rangle'$.
But, in the complex case, I can only get $\Re\langle x,y\rangle=\Re\langle x,y\rangle'$ with this argument.