What complications would arise living on a planet in a binary star system?

First up

You could never have a situation where there are two sun-like stars in the sky that rise and set like our sun, and never seem to change in size.

Only two situations are stable :--

And assume both stars are equal in brightness, and orbit in the same plane as the planet.

1) the two stars orbit each other at a fairly close distance -- less that the distance of our sun from earth -- then the planet orbits both stars.

In this situation there will be great variation in the amount of energy received by the planet. From when one star is at its closest approach the planet -- to when both stars are equidistant from the planet.

It is doubtful if life could evolve on such a planet.

2) The planet orbits one star at a similar distance as our earth is from the sun. The other star is at a great distance -- similar to Neptune's distance.

In this case life would be possible on the planet, and the 2nd star would contribute very little energy to the planet. it would appear as a rather bright star.

The problem in both cases, is that it is unlikely that any planet could form in a dual star solar system. Or at least, if a planet did form, it would be too far from either star to receive enough energy.