Generally, "adds another dimension" would mean a completely new factor has been added into consideration.
With a work of art, this could mean that it got richer: "switching the narrative focus to the the antagonist in the second act added another dimension". (And of course, puns abound when people make 3D films).
With a "situation" in which conflicting interests are at play, then it would indeed make things more complex. Imagine a plan to sell a public building to a private interest that was being opposed because it would stop the services currently available there. Now imagine that while the debate on that was going on, a new group started objecting to the environmental impact of the private interest's plans. That would add another dimension to the situation, because it's a completely new consideration in the argument about whether the sale should go ahead.
On the other hand, if there had been 10 people objecting at first on the grounds of services being stopped, and later there were 200, then the addition of more opposing views would complicate things - 200 are less likely to be convinced of a compromise solution than 10, for example - but it wouldn't be a new dimension.