As a lover of number theory, the condition $f(xy)=f(x)f(y)$ in group morphisms always reminds me of completely multiplicative functions. Since the natural numbers $\mathbb{N}$ do not form a group under multiplication, we will work instead in the non-zero rational numbers $\mathbb{Q}^*$ where the value of the multiplicative function at a rational number will be defined as

$$f\left(\frac{a}{b}\right)=\frac{f(a)}{f(b)}$$

which can be easily verified to be consistent under the multiplicativitiy hypothesis. This above representation means that we need to restrict ourselves to multiplicative functions that do not have $0$ under their image, but this isn't a huge loss. Collecting what we have above, every completely multiplicative function that does not have $0$ in its image induces a natural morphism $\mathbb{Q^*}\to\mathbb{C}^*$, and working backwards even morphism $\mathbb{Q^*}\to\mathbb{C}^*$ induces a natural multiplicative function $f$ that does not have $0$ in its image.

It feels very natural to me that the properties of morphisms $\mathbb{Q}^*\to\mathbb{C}^*$ have been studied extensively, and so I am wondering if there have been any papers published that utilize our knowledge about these morphisms to state new theorems about multiplicative functions.