Denote by $\mathbb{F}$ one of the set of (all) integer, rational, real or complex numbers.
We are looking for necessary and/or sufficient conditions for $\emptyset\neq C\subseteq \mathbb{F}$ such that $C=A+B$ for some non-singleton subsets $A,B$ of $\mathbb{F}$.
Some obtained results:
(1) Every co-set of a non-trivial additive subgroup of $\mathbb{F}$ enjoys the property.
(2) If $1+C\subseteq C$ (e.g., all additive sub-semigroups containing 1), then $C=\{0,1\}+C$ and so the propety holds.
(3) All intervals of real numbers ($\mathbb{F}=\mathbb{R}$) of the form $C=(0,c)$ satisfy the condition.
Any other idea (especially for finite sets)?
Thanks in advance