I was asked whether it was possible to produce a monic polynomial with integer coefficients, constant coefficient equal to $1$, having a real root $r > 1$ and a pair of complex roots with absolute value $r$, which are not $r$ times a root of unity. Bonus if the polynomial did not have roots of absolute value one. An answer (without the bonus) is:

$x^{12} - 4x^{11} + 76x^{10} + 156x^9 - 429x^8 - 2344x^7 + 856x^6 - 2344x^5 - 429x^4 + 156x^3 + 76x^2 - 4x + 1$.

I'd like an answer to the bonus question in the following strengthened form: Is there a unit $r$ in a number field such that $r$ has the same absolute value (bigger than one) at a real and a complex place (of $\mathbb{Q}(r)$ to avoid trivial answers) but no archimedian place where $r$ has absolute value $1$?