Conductivity of water and chlorine?
In a nutshell, water is very much not like semiconductors. Its conductivity is due to altogether different mechanism, and follows different trends.

Typical semiconductor is a rigid network of atoms held together with bonds. Nearly any "wrong" atom would provide either extra electrons or holes (that is, missing electrons). Holes and electrons are the charge carriers here. Being very small, they move fast, so you only need a tiny amount of those impurity atoms (say, a few ppm) to create significant conductivity.

Water is quite different. You can't have free electrons in water, to begin with; they will instantly react with something and become ions. Ditto for holes. Ions are the only allowed charge carriers in water. Ions are as big as atoms, and each carries a thick coat of water molecules stuck to it, so effectively they are even bigger than atoms. An ion would move in an electric field, but with great difficulty, much like a man pushing his way through a busy crowd. (Compared to it, the electrons would be like little sparrows darting overhead.) Therefore you need a lot of ions to create some real conductivity.

Things that produce ions in water (dissociate, as we say) are called electrolytes, and there is plenty of them around. For example, table salt dissociates into $\ce{Na+}$ and $\ce{Cl-}$; sulphuric acid dissociates into $\ce{H+}$ and $\ce{SO4^2-}$. Elemental chlorine ($\ce{Cl2}$), being a neutral molecule, does not dissociate; also, you can't add much of it, for it is a gas, and its solubility is not that great. Over time, it would react with water to produce certain compounds which do dissociate, so you'll have some conductivity after all, but that's a weird and inefficient way of achieving it. Using any soluble salt is much more straightforward.

The choice of particular electrolyte depends on your ultimate purpose.