Complexified tangent space

"The" book is right: $T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}$ can be identified with the complex linear vector space of $\mathbb C$-linear derivations $C^\infty_{M,p,\mathbb C}\to \mathbb C$.
Indeed, given the real derivation $v\in T_{\mathbb{R},p}(M)$ the elementary tensor $v\otimes z\in T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}$ acts on $f+ig\in C^\infty_{M,p,\mathbb C}$ (=germs of smooth complex-valued functions defined near $p$) by the formula $$ (v\otimes z) (f+ig) =z\cdot [v(f)+iv(g)]$$ This action is a $\mathbb C$-linear derivation $ C^\infty_{M,p,\mathbb C}\to \mathbb C $ and all such complex derivations are uniquely obtained from $ T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}$ .
To sum up in a formula: $$Der (C^\infty_{M,p,\mathbb C}\to \mathbb C)=T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}= Der_\mathbb R (C^\infty_{M,p,\mathbb R}\to \mathbb R) \otimes_\mathbb R \mathbb C $$

Did you notice that the the complex structure on $M$ is irrelevant?
No?
I'm not surprised: this fact is almost never mentioned in books or lectures.
The complex structure on $M$ gives rise to a canonical direct sum decomposition $T_{\mathbb{R},p}(M)\otimes_\mathbb{R}\mathbb{C}=T^{1,0}\oplus T^{0,1}$, where $T^{0,1}$ consists of derivations killing germs of holomorphic functions but that is another (long and interesting!) story.