Mainstream claims are improvements to:
Mental Health
Alertness
Muscle soreness
Weight loss
So what do people know cold water immersion and what has been your experience?
Cold showers and cold water swimming for mental health claims are older than you are.
In anxiety circles the vagal/mammalian diving response stuff got resurrected a couple of years ago and has been doing the rounds ever since around the internet. Usually you get people religiously quoting it (especially the face splashing) not quite understanding why, but they say it works.
In 2016 the BBC had a tv double parter called The Doctor who gave up drugs and that featured some of the research on cold water swimming.
The BBC has a follow up 2 years later https://www.bbc.co.uk/news/health-45487187 for someone who'd taken medication for 8 years and stuck in a rut and took up cold water swimming (supervised). The article explains the basic. On weight loss/depression people have been looking into how brown fat is activated in the body, as well as the stress response and cross-adaptation.
Alertness is one that's talked about a lot, it's quite a hard one to quantify. You could argue that going into parks for walks/cycles with birdsong and trees makes you more 'alert'.
For me cold showers don't work, they make particular muscles even stiffer. That one's not particularly unheard of if you seek enough experiences.
At a local park cold water swimming during covid has gone from virtually zero usage to daily swimmers (sadly the water isn't tested so it can't be verified that it's relatively safe).