Complexity theory uses a large number of unproven conjectures. There are several hardness conjectures in David Johnson's NP-Completeness Column 25. What are the other major conjectures not mentioned in the above article? Did we achieve some progress towards proving one of these conjectures? Which conjecture do you think would require completely different techniques from the currently known ones?
The unique games conjecture has recently been one of the most fruitful assumptions for proving good lower bounds, although it is more controversial than most assumptions about complexity class separation. See On the Unique Games Conjecture for a list of consequences.
This isn't mentioned in the article, but the exponential time hypothesis is very useful for proving exponential lower bounds on the running time of hard problems.
The polynomial hierarchy is infinite. This implies other conjectures that are often used, like NP is not in coAM, NP is not equal to coNP, etc.