Implement the random Fibonacci sequence
05AB1E, 8 7 bytes
λ₂D(‚Ω+

-1 byte thanks to @ovs.
Prints the infinite sequence.
Try it online.
Explanation:

λ        # Create a recursive environment to output the infinite sequence,
         # implicitly starting at a(0)=1
         #  (push a(n-1) implicitly)
 ₂       #  Push a(n-2) (NOTE: all negative a(n) are 0, so a(-1)=0)
  D      #  Duplicate a(n-2)
   (     #  Negate the copy: -a(n-2)
    ‚    #  Pair them together: [a(n-2), -a(n-2)]
     Ω   #  Pop and push a random item
      +  #  And add it to the a(n-1)
         # (after which the infinite list is output implicitly)