Is the fluctuation pattern of the Cosmic Microwave Background (CMB) identical in every location of the universe? I know that the Cosmic Microwave Background (CMB) is the leftover radiation from the "surface of last scattering". 
Let's say an alien civilisation lives on andromeda or further away. Would they see the same pattern in the fluctuation of the CMB?
If no, how fast does it change?
 A: There are other factors too, which also contribute to the pattern being at least slightly  different in different parts of the Universe. 


*

*Gravitational lensing.

*The interaction of CMB photons with high-energy electrons, in something called the Sunyaev–Zel'dovich effect.
Both of these depend on the specifics of what material comes between the detector and the path of the incoming CMB photon, along its entire path. This will of course vary from place to place in the Universe (but relatively slowly, on a human scale).
See also this related question.
A: You are correct, the surface of last scattering is different for different observers. Its distance is approximately 14,000 Mpc. 
For a given feature, The change in angle from one position to another is given by parallax, and for small angles is approximately $\Delta \theta=d/R$, where R is the distance to the surface and d the distance between vantage points. andromeda is at aprox. 780,000 pc, thus they will see our features shifted (and new ones included) by an angle of .0035 arcsec. For a 1deg shift, you need to travel  25 Mpc
