Following Can a real power series have an uncountable number of real roots? and this essentially equivalent question about a sort of linear independence of powers of a real function, the natural thing to ask is the following.
Let $a_n \in \mathbb C$ be a sequence of complex numbers, not all zero, and consider the power series $f(z) = \sum_{n=0}^\infty a_n z^n$. Can $f$ converge to $0$ on an uncountable subset $S \subseteq \mathbb C$?
By the following argument, all but a countable number would have to lie on the boundary of the disk of convergence $D$ of $f$. Indeed, $D$ is a countable union of compact sets $K_n$, and if $S_n = S \cap K_n$ were infinite then it would have an accumulation point inside $K_n \subset D$, hence by the identity theorem $f = 0$. Therefore $S \cap D = \bigcup_n S_n$ is countable.
