Axiom, 50 bytes

g(n:INT):INT==(n=0 or n=1=>n;n rem 2=0=>g(n/-2);0)

exercises

(70) -> n:=-10000;repeat(if g(n)=1 then output n; n>10000=>break;n:=n+1)
   - 8192
   - 2048
   - 512
   - 128
   - 32
   - 8
   - 2
   1
   4
   16
   64
   256
   1024
   4096

Axiom, 50 bytes

g(n:INT):INT==(n=0 or n=1=>n;n rem 2=0=>g(n/-2);0)

It would return 0 if n is not power of (-2), else return 1; exercises

(70) -> n:=-10000;repeat(if g(n)=1 then output n; n>10000=>break;n:=n+1)
   - 8192
   - 2048
   - 512
   - 128
   - 32
   - 8
   - 2
   1
   4
   16
   64
   256
   1024
   4096