Okay I have been working under the assumption that this is "obvious" for a while now, but it started to bug me and now I'm fumbling to prove it.
Suppose $X$ is a normed linear space (possibly infinite dimensional). Let $\mathcal{B}$ be a Hamel basis of $X$. Fix $b \in \mathcal{B}$. Is the coordinate projection $P_b : X \to \mathbb{C}$ or $\mathbb{R}$ defined by $P_b(b) = 1$ and $P_b(x)=0$ for $x \notin \mathrm{span}(\{b\})$ continuous?