By Gelfand-duality with $1$, the set $\{pt\}$ is homeomorphic to $\mathrm{Spec}(C(\{pt\})) \cong \mathrm{Spec}(\mathbb{C})$, the set of nonzero *-homomorphisms $\mathbb{C} \rightarrow \mathbb{C}$ with the pointwise convergence topology.
In particular, there can be only one * -homomorphism $\mathbb{C} \rightarrow \mathbb{C}$. But why can't I take both $id$ and complex conjugation (*) itself?
By definition $^*$-homomorphism is a linear map, but complex conjugation is antilinear