I have a Healer's Hawk, and I mutate Huntmaster Liger on top of it, the mutate ability triggers, and goes on the stack. If I am thinking about it correctly, if I held priority, could I cast Cloudshift to "separate" the creatures? Would the mutate ability on the stack still work? And if it did, is the reentering Huntmaster Liger considered "other creature"?
With Cloudshift, if the creature involved is a merged creature, all its components go to exile, then you return those components that are cards "to the battlefield under your control". The components will enter the battlefield as separate permanents rather than being merged. In particular:
Huntmaster Liger's "Whenever this creature mutates..." ability won't trigger this way.
The components that enter the battlefield will be different from the merged creature that left it as far as the game is concerned (C.R. 400.7h).
Here, the effect of Cloudshift "can find the new object that a merged permanent becomes as it leaves the battlefield, [so] it finds all" of those components; if "that effect causes actions to be taken upon those [components], the same actions are taken upon each of them", to the extent possible (C.R. 721.3c, 721.3, 400.7h, 400.2, 609.3). (Note in particular that a token component of the permanent will not return to the battlefield this way [C.R. 111.8]).
Yes, you can do that. The two cards will return as entirely new creatures, and thus wil be included in the "other creatures". The mutate trigger will use last known information to determine how often the no longer existing creature had mutated (which is once in this scenario) and give the bonus accordingly.
Here, the effect of Cloudshift "can find the new object that a merged permanent becomes as it leaves the battlefield, [so] it finds all" of those components; if "that effect causes actions to be taken upon those [components], the same actions are taken upon each of them", to the extent possible (C.R. 721.3c, 721.3, 400.7h, 400.2, 609.3). (Note in particular that a token component of the permanent will not return to the battlefield this way [C.R. 111.8]).
See also this thread, this thread, and this thread.
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