That is basically how the common law functions today. Harm people and you are personally punished. Harm property and you must pay for that property. Criminal law and civil law, crimes and torts.
*Of course the "property" in the original were actually slaves, but the principal holds.
200. If a man knock out the teeth of his equal,
his teeth shall be knocked out.
201. If he knock out the teeth of a freed man,
he shall pay one-third of a gold mina.
202. If any one strike the body of a man higher in rank than he,
he shall receive sixty blows with an ox-whip in public.
203. If a free-born man strike the body of another free-born man or equal rank,
he shall pay one gold mina.
204. If a freed man strike the body of another freed man,
he shall pay ten shekels in money.
Here are a couple more interesting clauses:
108. If a tavern-keeper (feminine) does not accept corn according to gross weight
in payment of drink, but takes money, and the price of the drink is less
than that of the corn, she shall be convicted and thrown into the water.
114. If a man have no claim on another for corn and money, and try to
demand it by force, he shall pay one-third of a mina of silver in every case.
And, again, there is no cultural continuity from mesopotamia to modern europe or america. Similarity between Hammurabi's Code and the English common law does not represent influence of one on the other, it represents parallel evolution. It is therefore no more and no less relevant to understanding the common law than, say, early Indian law, or 1700s Japanese law.