Things like this tend to be prone to
Sybil attacks. I'm not an expert but what I learned from comments like
this is that there are some very good reasons why Bitcoin is based on Proof-of-Work rather than trust. Not sure if this applies to your system.
Sybil is on my mind.
I plan a register of persons to prevent fake multiple identities. Assuming a core number of trusted users, every user would show up in person at a certain number of users in his area (partially to be selected by random choice, to prevent systematic collusion). The user would present some official documents and then get a uid based on a standardized id scheme, eg a hash on "Firstname Middleinitial Lastname-When-Born Birthplace Birthdate". To be able to act anonymously, the user would first logon using this id and then use this to generate secondary anonymous credentials for login. These will be produced in a manner that there will only be one credential for every user per time unit. Thus, the user is anonymous AND the risc of a large number of fake users is small. It is kind-of a decentral implementation of a PKI, similar to the web-of-trust in PGP (but preventing collusion attacks).
What you are describing requires strict identity verification, and thus also requires a central authority (to do said verifications) and lots of trust. In other words, it's nothing at all like Bitcoin. If you're going to require a central authority, don't bother with proofs of work or anything like that; just record balances in a database.
Absent some centralized mechanism, which is contrary to everything Bitcoin stands for, you have to realize that there is
absolutely no way to distinguish any sort of difference between one person running one hundred computers or one hundred people running one computer each. If you cannot tell who is running what there is no way any sort of "non-parallelizable proof of work" could possibly work.