Here is a characterization of continuous homomorphisms into $\mathbb C^{\ast}$ on a complex algebraic torus given by A. Borel in the Corvallis article Automorphic L-functions.
I don't understand where this characterization is coming from.
Consider the case where $T = \mathbb C^{\ast}$. Since $\mathbb C^{\ast} = S^1 \times (0,\infty)$, a continuous homomorphism (quasicharacter) of $\mathbb C^{\ast}$ is a combination of a character $S^1 \rightarrow S^1$, which is of the form $z \mapsto z^m$ for a unique integer $m$, and a quasicharacter $(0,\infty) \rightarrow \mathbb C^{\ast}$, which is of the form $x \mapsto x^s$ for a unique complex number $s$. Therefore, for any quasicharacter $c: T \rightarrow \mathbb C^{\ast}$, there are unique $m \in \mathbb Z$ and $s \in \mathbb C^{\ast}$ such that
$$c(z) = (\frac{z}{|z|})^m |z|^s = z^m|z|^{s-m}$$
As long as $z \not\in (-\infty,0)$, we can write $|z|^{2t} = z^t \bar{z}^t$ for all complex numbers $t$, and so
$$c(z) = z^m z^{(s/2) - (m/2)} \bar z^{(s/2) - (m/2)} = z^{(s/2) + (m/2)} \bar z^{(s/2) - (m/2)}$$
where $(s/2) - (m/2)$ and $(s/2) + (m/2)$ differ by an integer.
If $z \in (-\infty,0)$, then I don't think this procedure works. How do we account for $z$ negative, and generalize the arbitrary $T$?