Unfortunately this all means that the only actually interesting aspect of this note is the little fun fact that R, C, and any other Q-vector space of dimension |R| (like hermitian matrices, for instance) are all isomorphic.
While this is obvious when said like this, one can stump quite some mathematicians by asking them to prove that R and C are isomorphic as additive groups (which of course follows from the statement about Q-vector spaces). The point is that R and C are only very rarely thought of as Q-vector space so many would not think about this.
Follow up challenge/fun fact: show that the multiplicative group C{0} is isomorphic to the multiplicative group S1 = {z in C, |z|=1}