For which totally complex number fields $K$ with embeddings $\{ \sigma_1, \dots, \sigma_m\}$ do we have the equality:
$$ |\sigma_1(x)| = |\sigma_2(x)| = \dots = |\sigma_m(x)|, $$
for all $x \in K$ where $|\cdot|$ corresponds to the complex absolute value $|x| = (x\bar{x})^{1/2}$? In other words, which number fields have embeddings that do not affect the distance of a coordinate on an argand diagram?
Assuming your condition, we must have that every galois action acts via multiplication by some complex unit:
$|x| = |\sigma(x)| \implies \sigma(x) = u \cdot x$ for some complex unit $u$.
Assume $\sigma(x) = u \cdot x$, then $\sigma(x+1) = u\cdot x + 1 = w\cdot x + w$ for some other unit $w$. This would imply that $|x+1| = |u\cdot x + 1|$. This is absurd unless $u\cdot x = \bar{x}$ or $x$. Hence the galois group must consist only of the identity and the complex conjugate (only imaginary quadratic fields fit the condition).