Indy without Indy, I sincerely hope not!!
A spin-off series seems to definitely be a possibility, with Shia/Mutt being the obvious candidate. The big issue here already addressed is the time period. An adventure series set in the 60's-70's just wouldn't work, hence that's why the Indy films have all been in the 30's with the exception of KOTCS. The time period of KOTCS was also really only set to coincide with Ford's real age, and the passing of time of the character.
I for one would rather another film with Ford then a spin-off, but that's looking highly unlikely! For me it's either that, or end the series now! Call me crazy, but spin-offs would just seem to cheapen and tarnish the memory of this great movie series. The scarier prospect to me then spin offs are remakes!! I have no doubt that in the not too distant future we'll see some remakes of some real classics. I wouldn't put it past Hollywood to remake Raiders, Jaws, Star Wars, A Clockwork Orange, 2001: A Space Odyssey, and The Excorcist, just to name a few.