Let $\mathcal{B}$ be a normed linear space and $\{a_i\}_{i\in \Bbb N}$ a basis for $\mathcal{B}$. Consider $b= a_1+a_2$. It can see that $\Bbb C b = \cup_i (\Bbb C b \cap \Bbb C a_i)$. Is $$\Bbb C b \cap \Bbb C a_i\neq \emptyset$$ for every $i$?
If $\Bbb C b \cap \Bbb C a_i= \emptyset$, then $\Bbb C b=\emptyset$. Also, if $\Bbb C b \cap \Bbb C a_i\neq \emptyset$, then there are $\lambda,\mu_i$ such that $\lambda b=\mu_i a_i$. Thus $b=\frac{\mu_i}{\lambda} a_i$.
Where is my mistake?
We have $\mathrm{span}(b)=\Bbb Cb\subset\mathrm{span}(a_1,a_2)$, so, by linear independency of the basis, it intersects all other $\Bbb Ca_i$ trivially (in the zero vector) for $i>2$.
But it also has a trivial intersection with both $\Bbb Ca_1$ and $\Bbb Ca_2$, because else, as you conclude, $a_i$ would be parallel to $b$ (scalar multiples of each other).
This also shows that the hypothesis you started off is wrong.