Vector spaces: Is (the) scalar multiplication unique?

No. On any complex vector space $(V,+,\cdot)$ you can introduce a new scalar multiplication $*$ given by $z * v = \overline{z} \cdot v$ for all $z \in \mathbb{C}$ and $v \in V$.

More generally: If $(V,+,\cdot)$ is an $F$-vector space and $\phi \colon F \to F$ a field automorphism then $z * v = \phi(z) \cdot v$ defines a new scalar multiplication $*$.

PS: The scalar multiplication is unique if $F$ is a prime field, i.e. if $F = \mathbb{Q}$ or $F = \mathbb{F}_p$ with $p > 0$ prime. This follows because the action of $1 \in F$ is uniquely determined by the axioms of the scalar multiplicaton and each element in these fields is a multiple of $1$ (if $F = \mathbb{F}_p$) or can be written as a quotient of multiples of $1$ (if $F = \mathbb{Q}$).

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