Consider the plane $L$ spanned by a vector $V$ and its complex conjugate one $ \overline V$.
I'm trying to understand "why" a (real) vector $W \in L$ is expressed as $W=a(V+\overline V)+b(V- \overline V)/i = V'+ \overline V'$, with $a, b \in \mathbb R$...is it because in this way $V'=(a-bi)V$ and $\overline V'=(a+bi) \overline V$?
Well, first of all it is hopefully clear that we can write $W$ as a linear combination of $V_1 = V+\bar{V}$ and $V_2 = -i(V - \bar V),$ because they span $L,$ but we might need to use complex coefficients. So let's say $W$ is a real vector (ie $W = \bar W$) and that $W = aV_1 + b V_2$ for $a,b \in \mathbb C.$ We must have:
$$a V_1 + b V_2 = W = \bar W = \bar a \bar V_1 + \bar b \bar V_2$$
Now note, $\bar V_1 = V_1$ (as it just swaps the order of the sum) and $\bar V_2 = -\bar{i}(\bar V - V) = -i(V - \bar V) = V_2.$ Hence: $$a V_1 + b V_2 = \bar a V_1 + \bar b V_2$$ Since $V_1, V_2$ are a basis ($L$ is a plane) we only have one way to write a vector as a linear combination of them, so in particular $a = \bar a, b = \bar b$ and so $a,b \in \mathbb R.$