The Birkhoff–Kakutani theorem gives the necessary and sufficient conditions for a topological group $(G,\cdot)$ to be metrizable. It gives a construction of a metric $d$ on $G$, and proves that it is also left invariant, in the sense that $d(x,y)=d(g\cdot x,g\cdot y)$ for all $g,x,y\in G$.
In particular, it implies that there exists a metric $d$ on $\mathbb{C}^*$ such that it generates the same topology and $d(x,y)=d(zx,zy)$ for all $x,y,z\in\mathbb{C}^*$, but it doesn't give an explicit example.
How would such a metric look?
I tried doing this for $(\mathbb{R}_+,\cdot)$ by defining $d(x,y):=\left\vert\log(x)-\log(y)\right\vert$, and since the map $\log:\mathbb{R}_+\rightarrow\mathbb{R}$ is both a group isomorphism and a homeomorphism, it is quite clear that it would work. However, I'm not so sure the same trick would work here since $\log:\mathbb{C}^*\rightarrow\mathbb{C}$ has different branches and everything that arrises from $z\mapsto e^z$ not being injective in $\mathbb{C}$.
Does this work?
Thanks in advance to anyone who answers.
The group $\mathbb{C}^*$ is the direct product of the groups $\mathbb{R}_+$ and the group $S^1$ of complex numbers of absolute value $1$. The Euclidean metric is already an invariant metric on $S^1$, so you can just combine it with your logarithmic metric on $\mathbb{R}_+$. Explicitly, you can define a metric $d$ on $\mathbb{C}^*$ by $$d(ru,sv)=|\log(r)-\log(s)|+|u-v|$$ where $r,s\in\mathbb{R}_+$ and $|u|=|v|=1$.
Or, more directly along the lines you suggest, you can get an invariant metric by defining $d(z,w)=|\log(z)-\log(w)|$ where the right-hand side means the smallest possible value ranging over all possible logarithms of $z$ and $w$. This amounts to taking the imaginary part of $\log(z)-\log(w)$ to be the arc length distance along the unit circle between $z/|z|$ and $w/|w|$.