How can there be anything "beyond" the CMB? Two things we take, for this purpose, to be axiomatic:


The CMB is the oldest thing we can observe directly
The cosmological red shift tells us how "old" something we are observing is


Yet we also have a third axiom:


There are some parts of the universe we can never observe because they are receding away from us at a superluminary speed.


But this surely also implies that we can see beyond the CMB if we see anything which has a red shift indicating an expansion speed very close to $c$.

This suggests axiom 2 is incorrect - so what should it be instead?
Yet we also have a third axiom:
  
  
  There are some parts of the universe we can never observe because they are receding away from us at a superluminary speed.
  


This is simply false, so of course it gets you in trouble if you insist on taking it as axiomatic. The average recession velocity goes luminal at redshift $z\approx 1.4$, while there are observed galaxies and objects at $z\sim 8$, e.g., Z8 GND 5296 dwarf galaxy and the GRB 090423 gamma ray burst. Additionally, there are at least some candidates for objects much farther than even that, possibly $z\sim 12$.

That means that other than the CMB itself, the most distant and most ancient object we might have observed hails from when the universe as around $370\,\mathrm{Myr}$ old, which is around a thousand times older than the recombination epoch when the universe first became transparent and hence when the cosmic background radiation was emitted. In terms of redshift, the CMB has $z\gtrsim 1100$ or so.


  But this surely also implies that we can see beyond the CMB if we see anything which has a red shift indicating an expansion speed very close to $c$.


Something at redshift $z\gtrsim 1100$ has a recession velocity of $v\gtrsim 3.2c$.


  This suggests axiom 2 is incorrect - so what should it be instead?


Since (3) says incorrect things, it's best to throw it out instead.