Don't be stealin' my boxen

One needs to be precise with one's terms. "Communist," used without adjective, covers only orthodox or mildly heretical Marxists; another term is necessary to describe practitioners of the many derivatives and widely diverging offshoots of Marxism.

On the other hand, many socialists adhere to non-Marxian forms and are even anti-communist.

And then you have the Marxist school of historiography.

I label myself a utopian socialist. May I call you a market fundamentalist if that is indeed how you self-identify? If not, how would you describe the unfettered-market-solves-everything ideology?

"Market fundamentalist" is a snarl phrase not used for self-description. I think that:

1. People who claim that "the unfettered market solves everything" are a pack of shysters who have risen to prominence due to their unique willingness to (a) tell the voters what they want to hear, and (b) do enough of what needs to be done that the economy stays out of "total shambles" territory.
2. The entire idea that "the market solves everything" rests on the ridiculous assumption that everything can be solved; a notion that has reared its head noticeably in only two places, religious delusions and socialism.
3. Most people branded "market fundamentalists" do not, in fact, believe that the market solves everything, but are instead guilty of believing that the market can solve even one thing.
4. Although governments ignore the laws of economics to their peril, a free market does not preclude a role for the State, which can be more effective in running certain economic sectors, such as health care.

I seem to have had much less experience with the term "market fundamentalist" than you, so I suppose I can't talk about its common usage with as much certainty.

If you think healthcare is better run by the state, you are definitely out of the realm of market fundamentalism. Anarcho-capitalists like Murray Rothbard and David Friedman literally do think markets can solve everything, though. If that's not market fundamentalism, I don't know what is. I'd say there are other types of political fundamentalism as well, e.g. Communism (for obvious reasons), environmental fundamentalism (like the plants' rights people, eco-terrorists, and voluntary human extinction movement), and Constitutional fundamentalism (strict constructionists).