APL, 21 characters

'01'[-(16/¯2)⊤-2⊥⍎¨⍞]

I used Dyalog APL for this, with ⎕IO set to 0, allowing us to index arrays starting at 0 rather than 1.

Explanation, from right to left:

• ⍞ gives us the user's input as a character vector.
• ⍎¨ applies the execute function (⍎) to each (¨) of the aforementioned characters, resulting in a vector of integer 1's and 0's.
• 2⊥ decodes the vector from base 2 into decimal.
• - negates the resulting decimal integer.
• (16/¯2)⊤ encodes the decimal integer into base ¯2 (negative 2). (16/¯2 replicates ¯2, 16 times, yielding 16 digits in our negabinary number.)
• - negates each element of our newly encoded number (before this, it consists of -1's and 0's), so that we can use it to index our character vector.
• '01'[ ... ] indexes the character array ('01') using the 0's and 1's of the negated negabinary vector. This is so we get a prettier output.

Example:

      '01'[-(16/¯2)⊤-2⊥⍎¨⍞]
10111010001
0001101011010001

Ruby, 32 31 characters

m=43690
'%b'%(gets.to_i(2)+m^m)

Uses the negabinary calculation shortcut.

GolfScript, 34 29 27 characters

n*~]2base{.2%\(-2/.}do;]-1%

A plain straigt-forward approach. It is quite interesting that the shortest version is the one which first converts to number and then back to base -2 (at least the shortest version I could find up to now). But the nice thing about this one is, that it contains almost 15% %.

Edit 1: For base 2 we can save one modulo operation and also join both loops.

Edit 2: I found an even shorter code to convert binary string to integer.

Haskell, 86 83 Bytes

import Data.Bits
import Data.Digits
c n|m<-0xAAAAAAAA=digits 2$xor(unDigits 2 n+m)m

Call using c and then an integer array for digits, e.g.

c [1,1]

PS: I'm new, did I submit this correctly?

EDIT: Saved some bytes thanks to Laikoni and also fixed some typos

EDIT2: Alternatively, c :: String -> String:

import Data.Bits
import Data.Digits
c n|m<-0xAAAAAAAA=concatMap show.digits 2$xor(unDigits 2(map(read.(:[]))n)+m)m

For 114 bytes (but you call it with a string: c "11")

Python (2.x), 77 characters

(not as short as the other solutions due to the need to manually switch base...) Should satisfy the requirements.

i=input();d=""
while i:i,r=i//-2,i%-2;i+=r<0;d+=`r+[0,2][r<0]`
print d[::-1]

Suggestions for further improvements are welcome!

Feed it with starting values like this: 0b1001001

JavaScript, 68 bytes

function(b){for(r='',n=parseInt(b,2);r=(n&1)+r,n>>=1;n=-n);return r}

Would be 52 bytes in ES6, but that post-dates the challenge:

b=>eval(`for(r='',n=0b${b};r=(n&1)+r,n>>=1;n=-n);r`)

Jelly, 4 bytes, language postdates challenge

Ḅb-2

Try it online!

Takes input, and produces output, as a list of digits.

Explanation

Ḅb-2
Ḅ     Convert binary to integer
 b-2  Convert integer to base -2

This is pretty much just a direct translation of the specification.

k, 17 bytes _noncompeting

Some of the features used probably postdate the challenge.

1_|2!{_.5+x%-2}\2/

Input is a list of 1's and 0's, and output is also a list of 1's and 0's.

PHP, 69 Bytes

for($i=bindec($argn);$i;$i+=$i%2&$c,$i>>=1,$c^=1)$r=($i%2).$r;echo$r;

Online Version

ES8, 54B

b=>eval`for(r='',n=0b${b};r=(n&1)+r,n>>=1;n=-n);r`

05AB1E, 4 bytes

C2(в

Try it online!

Japt, 4 bytes

Input as a binary string, output as a negabinary digit array.

Íì2n

Try it

Or, taking input as a binary digit array:

ì2JÉ

Try it