Is there such a thing as an 'Indie'/Alternative anime scene?
There's Voices of a Distant Star, which was "directed, written, produced, character designed, storyboarded, cinematographed, edited, and animated by Makoto Shinkai". Essentially an individual effort except for some voice acting that was done by his wife Mika Shinohara. The DVD release had to go through a manufacturer and distributor, of course but I think that's as close as Indie/Alternative as you'll get.

There aren't many anime like that, as anime TV series or movies are very expensive and time consuming to produce. This is why even small anime studios often rely on production and backing from larger companies (like TV studios). Instances like Voices of a Distant Star is rare because it's short, and probably made no money.