The only cases I've ever found it necessary to anneal were .50-70's cut down to .56-56 Spencer Central Fire, and that's because you cut off the annealed portion when you form the Spencer cases.
When annealing brass, it must be heated until it glows, and then the anneal must be quenched in by dropping it into water. That is the opposite of steel, which is hardened by quenching after heating above the eutectic point.
My technique is to fill a used pill bottle with water and maybe put a bullet in the bottle (to keep it from floating). Then I set the pill bottle in a pot and fill the pot until a cartridge placed on top of the pill bottle is covered with water at least half its length. I place the whole shebang on a plastic lazy susan, so I can spin the pot while I heat the neck of the case with a propane torch. When the case looks hot enough, I knock the case off the pill bottle into the water to quench.
Nice thing about .44-40, especially Winchester brass, is the thin walls of the case don't really require annealing. I don't know what the life of a modern .44-40 case made by Winchester is...because I've shot mine at least 20 times, and the only losses are the ones that wind up burried in tall grass, or occasionally if my Rossi '92 catches the mouth of the case and causes so sever a "pitcher mouth" that I can'tt rebend it straight. (Fixed the problem by stoning the corners of the extractor groove in the breech end of the barrel.)
For reference purposes, .44 Magnum brass that is used for SASS-legal loads may be termed ".44 Extra Long Russian".
The logic is that it was common for a longer version of a cartridge to be called "Long". For example, the .44 Special should be called .44 Long Russian. It follows, therefore, that the .44 magnum should be called ".44 Extra Long Russian", except when loaded to modern .44 Mangle-em levels.