Hello,
a couple hours ago, Cisco released an update to the AnyConnect Secure Mobility Client addressing a specific installation problem (Cisco bug CSCuv03446) with OS X El Capitan.
Version numbers are 3.1.10010 or 4.1.04011 and should fix the problems mentioned at several different places here in the developer forums.
Talk to your organization's networking guys, ask them to install the updated versions on the VPN headends (e.g. ASA) and they should get to users automatically. In case your network admins need additional motivation, tell them that the update also includes support for compatibility with the new version of that unfortunately widespread desktop OS from that little northwest American manufacturer.
Please send feedback (i.e., more "interesting" bugs that you experience) with AnyConnect and El Capitan directly to "ac-mobile-feedback@cisco.com". One well-known bug with regard to unknown server certificates is still in there (bug ID CSCuv10340).
Not employed by Cisco, just relaying the news.
Cheers,
Christoph