Would an iPhone purchased in the US work in Algeria?

If you purchase the phone directly from an Apple store or Apple.com and ask for an unlocked phone it will work in Algeria, but if it ever needs to be repaired, either in or out of warranty, it will have to be returned to the country where it was originally purchased.

As long as you get the "SIM free" version at an Apple Store you should have no issues. But the usual caveats about getting official repair and/or warranty service from the Apple repair ecosystem apply. They reserve the right to refuse repair unless it's brought back to the country of original purchase. Granted a lot of the repair issue is because Apple will only replace entire devices with identical ones for many issues such as something defective on the logic board. If it's a bad display or replacing a battery is another issue and those are completely separate/universal parts.