Does Buddhism propose the possibility of any state of consciousness which feels better than neutral?
Many buddhist teachings I have heard so far (mostly by Alan Watts) seem to point to an "ideal" state of consciousness which feels neutral or maybe even not at all, as there is no conscious.
Does Buddhism claim to have some path to "eternal happiness" and more than "non-suffering"?
That's a very common misunderstanding of what Buddhism is after. Western authors (especially some of the early dabblers in Buddhism like Watts) had a very difficult time expressing Buddhist ideas in the parlance with which they and their audience would be familiar. Consistently, concepts like "voidness" (e.g. shunyata) get mistaken for ideas like nihilism, inertness, or Stoicism. You'd half expect the Buddhist ideal to more resemble Mr. Spock than Shakyamuni! I think this is even further compounded by the Pali and Sanskrit languages which often use prefixed negations to express positive qualities (e.g. ahimsa or non-harming). Based on that linguistic trope, it's very easy to think that Buddhism is only interested in a state in which negative qualities are absent.
Not to get caught in the weeds over what Buddhism is after (and you might want to stay away from words like "eternal" on a Buddhism Q&A site!), but let me assure your that liberation is anything but neutral. There are positive states like the seven factors of enlightenment (of which "joy" is one!), the compassion, equanimity, etc. of the Brahmaviharas, the sweetness of meditative absorption, among many, many others that are of paramount importance and have to be cultivated with just as much zeal as one uses to overcome their negative views and behaviors.
The Dhamma is "Akaliko": without time. It points to that outside Time and Space. Time and Space are ways the mind works, just as 'turning' is the way a wheel works. So the Dhamma points to that which is indescribable by the mind. It is not-mind. There is no 'eternal' there. There is no 'infinite' there.
Buddhism teaches "non-suffering". The whole path is about abandoning suffering & its causes therefore to have any other motivation (such as to find 'happiness') will not actually work.
This said, to not discourage people, the Dhammapada does explicitly provide the assurance that: "Nirvana is the highest happiness".
The result of "non-suffering" is a bliss the ordinary person who has had only ordinary experiences cannot imagine.
However, to imagine what Nirvana might be like, imagine the feeling of relief that occurs when you urgently need to & do urinate. This is a very blissful feeling from the release from suffering.
Even the idea of "neutral" is better than ordinary happiness. However, when the ordinary person thinks about "neutral", they think about depression rather than lucid sparkling clear purity.
Buddha taught everything, included better feeling consciousness, is suffering (dukkha).
Those teaching disclaim "consciousness which feels better".
But the the path (magga), to abandon that whole suffering, has arise with better feeling, in every step.
So someone think wrong that "Buddhism lead to eternal happiness". Because if buddha taught about "consciousness which feels better", so he shouldn't taught that "better feeling consciousness, is suffering".
P.S. In commentary said that magga happy feeling and phala happy feeling are suffering, too. But tanha can't desire them, so the practitioners have not to vipassana in magga and phala.
