Welcome to Atlantis, the second level of the Atlantis section from Tomb Raider 1.
In the area with the Doppelganger you have to dump the Doppelganger into the lava in order to be able to proceed with the level. I think it's assumed that the Doppelganger is killed as a result of this, although I'm not so sure after putting this video together.
In this video I manage to shoot at the Doppelganger after dumping it into the lava. This results in me taking damage and being killed. I'm not sure if the Doppelganger was still alive and shot back at me resulting in me being killed. Or if the Doppelganger was dead but the option to shoot the Doppelganger remained and the game is programmed so that shooting the Doppelganger harms you regardless of whether the Doppelganger is shooting back at you.
I show you the footage of myself shooting the Doppelganger after dumping it into the lava so that you can see it in action. I hope that you will find this useful and maybe come to conclusions as to whether the Doppelganger was dead or alive and whether the Doppelganger was shooting back at me or not.
Edit: Upon investigating further I've come to realise that the Doppelganger does still continue to shoot even after being dumped in the lava. I tried this again and stood in a position where I had a clear view of the Doppelganger and fired my guns (not at the Doppelganger) and I saw the Doppelganger shooting. I haven't added this to this video or done a separate video for it but wanted to inform you of this extra detail here in the description.
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