Today in a talk with a friend of mine I had an idea of extending cellular automatons to transfinite working time. I know it has already been considered, but, as far as I can tell, GoL extended to infinite numbers of steps hasn't been considered. At limit stages one would take limsup of value of the cell at earlier times. One can try to compare it to Hamkins-Kidder ITTMs: notions of writable reals and clockable ordinals could be extended fairly easily (e.g. configurations and stabilization times reachable from finite initial seed). What I am interested in is computational power of these machines. It is really unclear how one might simulate an ITTM in such system, because we can't guarantee if some operation will not repeat infinitely many times, and that can really mess up our system (for example, simple blinker becomes unstable, and it doesn't stabilize until stage $\omega5+25$).

My question is: has infinite time GoL ever been considered, and if so, has it been proven that it is ITTM-computably universal?