Homomorphisms from $(\Bbb Q,+)$ to $(\Bbb Q,+)$ and from $(\Bbb Q^*,\cdot)$ to $(\Bbb Q^*,\cdot)$

There are lot more automorphisms of $\Bbb Q^*$. By the fundamental theorem of arithmetic, $\Bbb Q^*\cong (\Bbb Z/ 2\Bbb Z)\times\bigoplus_p\Bbb Z$ where the $p$ range over all primes. The automorphism group of $\Bbb Q^*$ is uncountable, as for instance we can consider the maps fixing $-1$ and taking $p$ to $\epsilon_p p$ with $\epsilon_p=\pm1$.