Suppose I have a real vector space $V$ and I would like to extend the scalar multiplication in such a way that I obtain a complex vector space. It is not difficult to see that doing so is equivalent to fixing some linear map $U : V \to V$ satisfying $U^2 = -1$ and then defining $(a + ib)x = ax + bUx$ for $a+bi \in \mathbb{C}$ and $x \in V$. The next natural question is "when does such a map $U$ exist"? The answer is "always", provided $V$ is even-dimensional or infinite dimensional. To see this, take a basis for $V$ and split it into two collections of equal cardinality $(x_i)_{i \in I}$ and $(y_i)_{i \in I}$. Then define $Ux_i = y_i$, $Uy_i = -x_i$ and extend linearly.
But I don't have any idea how to do this and basically how to write all those proofs?