A frequent issue faced by divorced couple is when it is "appropriate" to introduce a significant other to the children, and if you are a spouse receiving alimony, the impact that "cohabitation" may have on support.
With regard to the parenting aspect of this question, it was recently addressed in the unpublished decision of Mantle v. Mantle. In Mantle, the Court was faced with whether to enforce a “DeVita” restraint, which is utilized to prevent the amount of exposure a divorcing parent may permit a child to have with the parent’s new girlfriend/boyfriend. Ultimately, the Trial Court in Mantle confirmed that exposing a child to a new dating partner, or even allowing a dating partner to stay overnight, is not per se inappropriate and contrary to child’s welfare and best interests. However, this came with some significant caveats, and the understanding that the dating partner has acted, and will continue to act, appropriately around the child. First, divorce is not like a Band-Aid, in which you can simple tear-off the old relationship and replace it with a new one. The Court was cognizant that children should not be thrust into an emotionally overloading situation overnight, and that a gradual transition and introduction over a reasonable period was appropriate so long as the restraints are reasonable and sensible. Obviously, notwithstanding the Court’s determination that exposing a child to a new dating partner was not per se inappropriate, it is the best interests of the children that always remain paramount. Should a new dating partner act inappropriately around a child, or even go so far as to verbally harass, torment or abuse the child physically, the Courts are always available to entertain necessary restrictions to protect the child’s interests. Although many individuals going through divorce think that they will never date or marry again, the Court acknowledged that human nature proves otherwise. People seek companionship, and it’s likely that divorcing spouses will enter into new relationships with new partners. Although the timing may vary, the Court noted that a new relationship is a natural and constructive step into moving on from one of the more difficult times in an individual’s life.

As to the impact of cohabitation on an existing support aware, this is governed by New Jersey’s alimony statute, N.J.S.A. 2A:34-23, which was recently amended on September 10, 2014. While it is clear that the new alimony statute makes it easier to establish cohabitation, questions linger as to how the new statute impacts cohabitation. One example is whether the old or new alimony statute applies to cohabitation that occurred before the new statute became effective.
Our New Jersey Appellate Division, in an unpublished decision, Schlumpf v. Schlumpf, ruled that where the cohabitation arose prior to the statute’s enactment on September 10, 2014, the former law on cohabitation will govern. The former alimony statute provided that cohabitation alone is insufficient to terminate alimony, meaning that cohabitation must be accompanied by a substantial economic benefit resulting from the relationship. A substantial economic benefit could refer to a having a joint bank account, sharing living expenses, or any other duties that are commonly associated with marriage.

No one can tell you who to live with or not live with, ,and it can affect your alimony in the sense that it could end it.

There are no laws that prohibit you from living with somebody, and by somebody I take it you mean a paramour. Doing so could certainly affect your alimony, up to and including termination of alimony.