For a 2D game, I can picture it working something like this...
It would well... work like a normal 2D sonic game,
though as your running with Sonic, Shadow will be right running right behind him.
When you press a button, they would then switch places. (They would be like a tag team.)
Sonic and Shadow would each have different abilities. (And some of the same of course.)
Sonic would have his homing attack,
and his bounce attack from Sonic adventure 2,
maybe even make him able to run up, or on the side of walls, etc.
Shadow would have the homing attack as well,
and he would also have teleportation, but not to be over powered,
I would limit it so that he could only warp directly left or right, and not to far, but still a good length.
(He would be able warp past enemies, and thin walls.))
Maybe give him the ability to use his chaos spear attack, and etc.
When playing the since Sonic is being followed by Shadow. (Or vice versa.)
You would be able to preform all sorts of combos.
For example:
Your controlling Sonic and you see an enemy, so you preform a homing attack,
while sonic is preforming it, then you can switch your control to shadow,
and warp past the enemy and throw a chaos spear, doing some combo of some kind.
(Perhaps put everything except you in slow when switching characters for a combo,
or well... have the follower preform combos with you by himself. (That'll probably call for an AI of some sort.))
Though yes, Sonic can Shadow can preform some several combat combos,
such as Shadow grabbing Sonic by the hands and swinging him around as Sonic is spinning feet at high speeds, etc
(The neat series Mario Bros Z does have Sonic and Shadow preforming several awesome combos
if your looking for combo attack ideas, but I am not sure how many of them would work for a game though.)
To show team work when traveling, (and to make things a little more realistic... (for a video game that is.))
when Sonic runs up walls with Shadow, he should probably grab Shadow by the wrist or hand,
since Shadow can't really run up walls with his style of running... or those hover shoes. (I think...)
And when Shadow teleports with Sonic,
he should probably grab Sonic by the wrist or hand in order for Sonic to warp with him.
(Since well... Sonic can't teleport.)
Oh, and of course the movement needs to flow really well,
teleporting, attacking, etc, if preformed right shouldn't slow you down.
Like a Super Mario bros speed run.
(Cough cough, and yes... I would recommend a Chao Garden as a bonus feature of course.

)
Shrug,
these are just some thoughts I am throwing out at the top of my head.