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No, not necessarily. For example, you can take $K\neq \mathbb{Q}$ to be totally real, and take $L$ to be a quadratic CM extension of $K$ whose unit group is equal to that of $K$ (e.g. make sure that $L/K$ ramifies at some place that is not above $2\infty$). Then ${\rm Norm}(U_L) = U_K^2$, and the unit index is $2^{{\rm rk}U_K+1}>2$.