"I lost the case"
No you didn't. The state lost the case. Victims of crimes are not parties to, and have no legal interest in, the resulting criminal prosecutions.
"Is the process for complaining about a DA the same as against any attorney?"
You can complain to his/her superior (I'm assuming this is an assistant DA rather than the DA, who is (I assume) an elected official). You can also complain to the state bar, but, since you were not a party to the action or a client of the DA, I doubt the state bar would do anything.