Tensor product of $\mathbb R$ and $\mathbb C$ over $\mathbb R$. $$ \mathbb{C} \otimes_{\mathbb{R}} \mathbb{R} = \;? $$
I guess this guy is just $\mathbb{C}$, is this correct?

Yes. In fact, for any commutative ring $R$ with module $M$, you can say this about the tensor product with free modules: $$M\otimes_R(\oplus_{i=1}^n R)\cong\oplus_{i=1}^n M$$
