Actually, you don't even need new DNA letter to do that. DNA codons can encode 64 different amino acids (63, as one codon must encode the end of sequence), but only 20 amino acids are actually used.
Adding another amino acid is theoretically possible, but this would require rewriting the whole DNA to reencode, say leucine from CTG to another codon to assign CTG to some other acid.
More likely, the new bases can be used as a higher-density data storage medium for those folks interested in making biological data stores.
biochemistries listed here are denoted as bonding atom/solvent pairs.
I think for now we are still a ways off from this being interesting at the protein level. I would think you would need new tRNAs to recognize the new bases in-order to really utilize them at a protein level, and those tRNAs would need to bind to different amino acids than we currently have for there to be any new protein function that we can't already accomplish with ATCG.
That being said, you can still do a lot of interesting stuff with nucleic acids like DNA and RNA, more and more research these days show they can do more than just encode information for proteins.
Possibly the RNA could have some secondary function in the folding of the protein or as a complex inside it...
Edit: spelling