Chillingworth asks Hester to promise that she will never betray his identity to anyone. In the interest of fairness, since she has chosen to keep the name of her lover secret, Hester must likewise honor his desire for anonymity. Chillingworth orders Hester not to betray him by word, sign, or look.
When Hester questions Chillingworth about his motives, he answers cryptically. First, he maintains that he prefers to stay anonymous so that he will not "encounter the dishonor that besmirches the husband of a faithless woman." Then, he admits that there may be other reasons; however, he asserts that he need not share those reasons with her. It is enough that it is his "purpose to live and die unknown."
Hester agrees to do what Chillingworth says, only because he threatens repercussions against Reverend Dimmesdale (Hester's lover) if she ignores his injunction not to expose him (Chillingworth): "Shouldst thou fail me in this, beware! His fame, his position, his life will be in my hands. Beware!"
Essentially, Chillingworth threatens to hurt Hester's lover. This is a little disingenuous of him. Remember that, earlier, he had promised he would never hurt the man who got Hester pregnant:
Yet fear not for him! Think not that I shall interfere with Heaven's own method of retribution, or, to my own loss, betray him to the gripe of human law. Neither do thou imagine that I shall contrive aught against his life; no, nor against his fame, if as I judge, he be a man of fair repute. Let him live! Let him hide himself in outward honor, if he may! Not the less he shall be mine!"
Yet, now, when he wants to protect his reputation, Chillingworth has no problems with threatening harm upon Hester's lover. In this chapter, Chillingworth's disturbing threat foreshadows his later sinister involvement in Reverend Dimmesdale's life.