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They fight in Empire City, but there are no people around so Alex can't feed. Kessler also can't go back in time and kill Alex pre-virus...
Who wins?

Not really sure how Cole1 attacks will effect Alex, but when he rests, as he does quite alot, one hit will end him.

You don't think the resting is merely a game mechanic? If he would never rest, he wouldn't be the easiest to defeat.

Think it was played out on his bad health and 'being weak' as he says at the half way point im currently playing at. Old timer needs to take a breather.

It's kinda hard to determine due to the depths of Kessler's power never being fully shown... We know he was developing new powers as time passed, and the time manip may have been his strongest ones, but he had gone WAY beyond simply throwing lightning around like Cole was, he was summoning those spectral clones, earth moving, teleporting around, and had strength & speed over Cole, who is a powerhouse in his own right... plus Kessler was holding off quite a lot during his fight with Cole, for obvious reasons.

That asside, Kessler was still no match for the Beast though, which tells me there is an unquantified limit in there somewhere...

I think Kessler wins. He has 80 years of experience under his belt with all of his powers. Being no match for The Beast is irrelevant as he only failed because he tried to fight it too late, meaning it was monstrously powerful by that time, and he was too late.