I think JMAP http://jmap.io seems to make sense for webmail, apparently they (or at lest someone) is adding support for JMAP in cyprus3.0
Its only used my FastMail (which has a very nice client) but isn't widely supported enough to warrant the major recode that would need to happen to support it.
I realize this is old, but still relevant.
Does roundcube have any plans to add JMAP support?
There aren't any concrete plans that I'm aware of. There is an open ticket https://github.com/roundcube/roundcubemail/issues/5918 for adding support but I don't think there is any ongoing work.
Roundcube next was going to support JAMP but that project came to an end. It would require a lot of rework in the Roundcube core to support JMAP so I wouldn't hold out hope for it anytime soon.