Is $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$? I'm trying to see if for several cases changing the ring in a tensor product affects the result or doesn't. Now I'm trying to prove  $\mathbb{C}\bigotimes_\mathbb{R}\mathbb{C}\simeq \mathbb{C}\bigotimes_{\mathbb{C}}\mathbb{C}$ if it's true, or to show why it isn't.
I've been unable to find an isomporphism between those two, but I don't know how would I proceed in order to show that there is no possible function that could define one.

There is an isomorphism of rings $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}$ (Hint: Use $\mathbb{C}=\mathbb{R}[x]/(x^2+1)$ and then CRT), but $\mathbb{C} \otimes_{\mathbb{C}} \mathbb{C} = \mathbb{C}$. So these are not isomorphic rings, since $\mathbb{C} \times \mathbb{C}$ has zero divisors for example. But they are isomorphic $\mathbb{Q}$-vector spaces, since the dimension is $c$ (continuum) in each case.
