Let $V$ be a real vector space. Using axioms of vector spaces you can prove that you only need to find out what $\imath \cdot v$ should be for $v\in V$. So let $ \phi(x) = \imath \cdot x $
Since $r_1 \cdot (r_2 \cdot v) = r_1r_2 \cdot v$ you get for $r_1 = r_2 = \imath$ the following equality must hold $ \phi(\phi(v)) = -v $ for all $v \in V$. Moreover, again using vector space axioms you can prove that $ \phi(a\cdot x + b \cdot y) = a\phi(x) + b\phi(y) $ for all $a,b \in \mathbb{R}$ and all vectors $x,y \in V$.
So existence of such a linear mapping $\phi: V \to V$ that squares to minus identity is equivalent to possibility of extending your real vector space structure to complex vector space structure.
The rest is up to you. Hint: Suppose such a $\phi$ exists. What can you say about the dimension of $V$?