The basic property you need is solubility. Due to the price of methanol-d4 relative to CDCl3, there is probably a good reason for using it (but not necessarily).
Using protic solvent, such as methanol-d4, that undergoes H/D exchange with NH, OH, etc. can helpfully simplify the proton spectrum relative to an aprotic solvent like CDCl3. Peak broadening from rotameric forms of Boc carbamates can also be significantly reduced in protic (or wet) solvents.
Chemical shifts are affected by solvent. Sometimes a spectrum run in e.g. CDCl3 might have several overlapping peaks that make assignment difficult. It may be the case that the overlap is much less in a different solvent. If you run yours in CDCl3 you will see a similar pattern of signals as reported in methanol-d4 (plus exchangeable OH/NH), but the shifts will be not be exactly the same.