If there were more parties, it would be harder for any one party to exert control. Majority would not be a matter of getting a few more seats for your party. Therefore, I agree with post 5 that alliances and negotiation would be necessary. There might even be more focus on the issues and lesson on the party line!
It would force the body to compromise, as now there is always one party in control of Congress. I like the idea, sort of like a Parliament, in that, with a large society like ours, minority beliefs and interests could still be heard, and not always ruled over by the majority. I also think Congress would get more done than it does now.
Multiple political parties in the American Congress would certainly open the door for numerous alliances and coalitions that currently are not workable. For example, the Blue Dog Democrats are conservatives who differ strongly with their party's national leadership. As a result, if they enter into alliances with like-minded Republicans, they are subject to the disapproval of the DNC, which means at election time they become less favored financially.
This is a great question as it forces us to consider the potential limitations of having a dominant two-party political system. Having more credible parties would, as #2 suggests, encourage working together and alliances, rather than the "trench warfare" between the Republicans and the Democrats. It would also hopefully encourage a healthier political system by giving choice and allowing party lines to be broken down.
The chief difference between the current two-party system and a multiparty system in Congress would be that the leadership positions in Congress would have to change. The Speaker of the House, who has significant influence over which bills make it out of committee and on to the House floor for debate and vote, is always a member of the majority party. In the Senate, it is the Senate Majority Leader who controls the flow of legislation from the committees to the Senate floor.
In a multiparty system, the different parties would also have to work harder to come to consensus with each other in order to assemble a majority vote on each bill. This would probably lead to more dialogue and compromise, but also slow down the legislative process for that dialogue to occur.
I think that one difference would be that there could exist a greater potential for alliances and coalitions formed between the parties. In the current system, even though there might not exist a huge practical difference between Democrats and Republicans, there is little chance of bipartisanship because of what it would cost politically. Both parties end up defining themselves in the most oppositional of terms. If we had multiple parties in our legislature, there could be a greater chance of collaboration and cooperation present because of simple math. If each party refused to work with the other, the lack of a sizable majority would play a significant role in stopping all legislation would compel legitimate attempts to reach across the aisles and force cooperation and coalitions to be formed. I think that this would be a significant difference in our congressional makeup with the legitimate presence of multiple parties.