Is this possible? How would this work?
The closest thing resembling a hash, or indeed a structure that can store data is nucleic acids, DNA, RNA and the like. Assuming you could engineer an organism to perform these tasks you can make some direct comparisons between Hashcash and a biological proof of work.
Instead of hash collisions you could match a DNA base sequence by chopping the DNA at random intervals (e.g. with restriction enzymes) - when you have the correct piece - say GATACCA you can make mRNA from this which codes for a polypeptide the cell can later secrete. Other cells could detect the presence of this polypeptide/protein and know that you had chopped your DNA up correctly. At this point they might take in the protein and combine it with another molecule creating a chain of polypeptides sort of like the bitcoin blockchain.
Somehow the protein needs to represent transaction data, perhaps encoded in the amino acid sequence but apart from that this could kinda work like bitcoin.
I hope I explained that well enough.
[1] http://en.wikipedia.org/wiki/Proof_of_work [2] http://en.wikipedia.org/wiki/Hashcash