Does flattr complement or compete with bitcoin?
Flattr is a social micropayments system. Bitcoin is a currency.

There is no reason the two could not be integrated together, pay with bitcoins and then share the bitcoins out using the same method as flattr.

In fact, I would go as far as saying this would be an excellent idea due to lack of transaction fees and friction for small amounts.

Flattr is the service, Bitcoin is the currency. They don't compete; they could certainly complement each other though, in my opinion.