[1] - https://plato.stanford.edu/entries/phenomenology/
[2] - https://plato.stanford.edu/entries/heidegger/
[3] - Wikipedia tends to be a pretty horrible source for reading about philosophy, so I don't like to link to it for this. The plato.stanford.edu site is better, at least for Analytics, not so much for Continentals. So I hesitate to link to it for this either. Better to just go to the primary source material, or take a course at one of the rare schools which focus on Continental thought, so you don't get the wrong idea by learing about Continental philosophy from Analytics who all too often see Continentals contemptuously as "not philosophy" or as just some version of Analytics, both of which miss the point.
[4] - Though some, particularly some Christian Existentialist thinkers have in fact interpreted Heidegger's thought from a Christian perspective. But that view, as most any other about Heidegger, really depends on who you ask. Many others interpret Heidegger's thought strictly non-theistically.
Consciousness might not be reducible to the material, or the material could all have a conscious component. There might be other things, like societies or biology, which are strongly emergent. Or maybe there is no material world. Maybe it's all mental. Perhaps there is a third substance, a neutral monism giving rise to mind and matter.
Maybe the noumena is beyond human categorization and perception. Perhaps our experience of the world is such because we are the animals we are, and not because we have access to reality in itself. Man is the measure and all that.
And on and on. There are various anti-realists debates. There is Humean skepticism of causality. None of these issues need invoke the supernatural, although they don't necessarily rule it out. David Chalmers can propose consciousness as an additional feature of nature which is nonphysical (tied to informationally rich processes like certain mental or computational states), without proposing any supernatural element.
First, how others perceive your words usually matters more than the precise definitions you have in mind. If I describe myself as believing in the supernatural, it has a lot of connotations I don't want.
Second, the possibility that consciousness (in some sense) could be built into physics themselves. So it doesn't contradict that the brain is made of subatomic particles, but the workings of the universe have more correlations built in that could work at different levels. Something I think is comparable is that many cosmologists think that the arrow of time, which events come before which, can be explained by a universe with initial conditions of extreme low entropy. So there's nothing within the universe that explains why the initial conditions were like that, but it's a real property of our universe.
Maybe consciousness could be an end state that the universe is progressing toward. Is that really so much weirder than working from an initial state of extremely low entropy?
If God exists, there's no reason there couldn't be new phyisics describing how his world works.
https://en.wikipedia.org/wiki/Vitalism
Either everything exists in the material world, or it doesn't. There's really no way of avoiding that binary. If you think there's something else, that is by definition supernatural.