By asking for complexity they want a number that they can compare with your next sprint to find your velocity as a team. They may also be trying to use it to add together your result with other teams to provide an over all estimate on when all the stories will be done.
I find the most important part of story pointing meetings are the discussions that come up. If everyone in the team is not able to feel confident suggesting a number then they don't fully understand the story and there needs to be more discussion. From the Agile Manifesto "Customer collaboration over contract negotiation". Rather than just specifying a contract written as a user story and saying the team failed if they don't do exactly what you want, it is always better to talk about what the customer really wants until you understand it.
To address your specific question about complexity vs effort, everyone seems to have a different opinion on this. Mountain Goat, who are the people who make the planning poker cards that have goats on the back of them and whose owner Mike Cohn is big in the Agile / Scrum world, say that it should be effort and not complexity.
I believe the point really should be the discussion but if you really need a number then just try to make it one that is independent on works on it and in what order the stories are worked on. The point it to get to a back log that is both priorities so you are working on them in the right order and sized so that the Project Manager can see roughly how many more you can fit in before some deadline. You should be able to stop at the end of any iteration and ship it with the features completed that had been started.