Consider a field automorphism $\sigma\in\textrm{Aut}(\mathbb C)$, and moreover consider the $\mathbb C$-scheme $p:\mathbb P^1_{\mathbb C}\longrightarrow\textrm{Spec}\,{\mathbb C}$ where $\mathbb P^1_{\mathbb C}=\textrm{Proj}\mathbb C[T_0,T_1]$.
We can define the $\mathbb C$-scheme $$\textrm{Spec}(\sigma)\circ p:\mathbb P^1_{\mathbb C}\longrightarrow\textrm{Spec}\,{\mathbb C}$$ that we indicate with ${\left(\mathbb P^1_{\mathbb C}\right)}^\sigma$. Now I have to find an isomorphism of $\mathbb C$-schemes (not only an isomorphism of schemes) between ${\left(\mathbb P^1_{\mathbb C}\right)}^\sigma$ and $\mathbb P^1_{\mathbb C}$. Do you have any idea about the construction of this isomorphism?
attempt of solution: Consider the induced isomorphism of $\mathbb C$-algebras $$\overline\sigma:\mathbb C[T_0,T_1]\longrightarrow\mathbb C[T_0,T_1]$$ where the rightmost $C[T_0,T_1]$ has the following structure of $\mathbb C$-algebra: $$\mathbb C\owns a\mapsto\sigma(a)\in \mathbb C[T_0,T_1]$$
In this way is well defined the map $$\textrm{Proj}(\overline\sigma):{\left(\mathbb P^1_\mathbb C\right)}^\sigma \longrightarrow\mathbb P^1_{\mathbb C}$$
Thanks in advance
Your solution is correct (and almost complete).
More abstractly, notice that there is an isomorphism of $\mathbb{C}$-schemes ${\mathbb{P}^1_{\mathbb{C}}}^\sigma = \mathbb{P}^1_{\mathbb{C}} \otimes_{\mathbb{C},\sigma^{-1}} \mathbb{C}$, where on the right we tensor with $\sigma^{-1} : \mathbb{C} \to \mathbb{C}$, and the $\mathbb{C}$-scheme structure is induced by the right tensor factor. (Of course, actually we take the fiber product with $\mathrm{Spec}(\mathbb{C}))$ etc.).
Now in general, if $f : A \to B$ is any homomorphism of commutative rings, then there is an isomorphism of $B$-schemes $\mathbb{P}^1_A \otimes_{A,f} B = \mathbb{P}^1_B$. Applying this to $\sigma^{-1} : \mathbb{C} \to \mathbb{C}$, we get $\mathbb{P}^1_{\mathbb{C}} \otimes_{\mathbb{C},\sigma^{-1}} \mathbb{C} \cong \mathbb{P}^1_{\mathbb{C}}$ where now the $\mathbb{C}$-scheme structure is the usual one.