Conway's Game of Points!
APL (Dyalog Extended), 34 32 bytes (SBCS)

Anonymous tacit prefix function. Requires ⎕IO←0 (zero-based indexing).
⌊/{⍺-1-⍸⌂life 0,∘⌽∘⍉⍣4⍸⍣¯1∧⍵-⍺}⊢

Try it online!
⌊/{…}⊢ call the following anonymous lambda with the smallest existing x and y coordinates as left argument (⍺) and the full list as right argument (⍵):
 ⍵-⍺ subtract smallest point from all points (scales so every number is non-negative)
 ∧ sort ascending
 ⍸⍣¯1 generate a Boolean matrix with 1s in those positions (lit. inverse of &quot;ɩndices of 1s&quot;)
 0…⍣4 repeat four times with zero as left argument:
  …∘⍉ transpose, then
   …∘⌽ mirror, then (this constitutes a 90° clockwise turn)
    …, concatenate zeros to the left edge
 ⌂life compute the next Game of Life generation
 ⍸ ɩndices of 1s
 1- subtract those from 1
 ⍺- subtract those from the offset of the lowest x and y coordinates