“I am apartment-hunting.” and “I’m looking for an apartment” are different?
They mean exactly the same thing 
You can also say “I am flat-hunting” or “I am flat hunting” which is British English and not used by everybody.
Here are some expressions:
I am house hunting
I am apartment hunting
I am flat hunting
I am job hunting
I am looking for a new house
I am looking for an apartment
I am looking for a flat
I am looking for a job
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Two more:
Police and US Marshals are on a manhunt for an escaped convict.
Women at college are on a man hunt for a husband to get their MRS degree.
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I’d say “manhunt” is being used a little figuratively in the second example.
ya think?