I think two discussion points emerging from Plath's "Lady Lazarus" would be its use of Holocaust imagery and its display of cruelty.
One of the poem's most dominant features is its use of the Holocaust. The imagery of the speaker's skin as a "bright Nazi lampshade" or her face as "fine Jew linen" are early examples of this in the poem. The linking between death and the Holocaust is a vital part of the poem. It might be a good discussion point to explore how the context of the Holocaust helps to enhance the poem's meaning. Discussion around this point could explore how Plath is trying to connect the historical experience of the Holocaust to the person who is enduring psychological struggle. The Holocaust is a significant element to the poem and a discussion about it could reveal unique elements to both the time period and Plath's use of it.
Part of the discussion regarding the Holocaust extends into the poem's understanding of cruelty. Plath brings out some very distinct examples of how human beings are cruel to one another. From the "peanut crunching crowd" to the "poke and stir" that accompanies the speaker at the end of the poem, barbaric behavior is a significant aspect in the poem. A discussion on this point could be whether this pain is deliberate or unintentional. There could also be examination as to whether this cruelty is from other people or from the speaker's own sense of self. When she says, "dying is an art" that she does "well," is this an example from external cruelty or from within? This is another discussion that emerges out of "Lady Lazarus."