There will probably always be money; a pure credit economy is unthinkable. Without money there is no price level, because the price level is defined as the average price of goods in terms of money.
A pure credit economy has money. It's defining characteristic is that all money is a liability of some issuer. It represents a debt of someone. Checkable deposits, for example, serve as media of exchange, and they are a debt to the banks that issue them. Privately-issued banknotes can serve as hand-to-hand currency. They serve as media of exchange, but they are credit money. They are debt to the issuing banks.
From the point of view of the firms and households making payments, these bank liabilities are assets that serve as media of exchange. They can quote prices in terms of them, receive payments of them, and then make payments with them.
Now, if the monetary liabilities of many banks are to be accepted at par, there needs to be some kind of clearing system between the banks. It is possible to have a settlement system where the quantity of the settlement medium demanded in equilibrium is zero. And so, it is arguable that there is no base money. All the money is credit money.
However, even if banks do hold balances of some settlement medium, that doesn't mean that it is necessarily "outside money," that is, a liability of no one. Suppose inter-bank clearings are handled by a private clearinghouse, and the clearinghouse creates (and destroys) balances by ordinary open market operations. It purchases and sells assets. The balances are liabilities to the clearinghouse. It is another type of debt.
Whether or not it is desirable, a pure credit money system is thinkable. There is money. Prices can be quoted in terms of that money. A price level can be calculated in terms of that money. It is even conceivable that the quantity of reserve balances created by a private clearinghouse would be adjusted to stabilize some other nominal value, like a growth path for nominal GDP. For example, if the clearinghouse was obligated to maintain index futures convertibility.
Now, if we consider the status quo, where an independent government agency issues base money, the only real question is whether the currency and reserve balances are best understood as a special type of debt or not. This very much depends on how serious is the commitment to the target for some nominal variable like inflation or a growth path of nominal GDP. A monetary authority organized on banking principles and subject to index future convertibility, using open market operations to change the quantity of base money, and so matching its issue of base money with financial assets, looks very similar to the private clearinghouse described above.
Is the current Fed's commitment to its inflation target enough? It is an empirical question. And Ashwin Parameswaran was correct about that.