Learn English – What’s the meaning of “with” in “Trembling with fear”

There's no notion of dying in that sentence.
You correctly understood that get done with something means to finish something, but it also adds a notion of relief as in

Finally! I'm done with that.

Jaspers even records a dream in which, during a tense conversation with some of Heidegger's critics, his friend suddenly approached and addressed him for the first time with the familiar du. The two then set off together, alone.

It is quite possible that the use of together and alone next to each other can cause difficulties interpreting this sentence, both to non-native and native speakers. I fall into the second group.

Alone basically means without others or by oneself. As Oxford puts it:

Having no one else present; on one’s own

In fact, as an adjective, alone often means solitary, in the company of one's self.

Whereas together basically means with someone or in another's company. As Oxford puts it:

With or in proximity to another person or people.

The native speaker in me wants to read together in its primary meaning and not, say, as at the same time.

Therefore when one reads this sentence one may very well pause and ask How can two people set off (depart) together (in one another's company), alone (by oneself)? This is the most natural reading of this sentence, yet it seems to defy standard word meaning.

A possibility is that since this occurs in a dream, the normal meanings of words do not apply. Another possibility is that the writer is using irony, saying that the two set off together but were so caught up in their thoughts that they might as well have been alone. (They were philosophers after all!) Another possibility is that the author has chosen his words poorly, that is, his sentence needs improvement. But a fourth possibility seems most likely, and depends on normal, straightforward meanings and even the collocation of alone and together.

Notice that in the dream there are others present in the scene, namely, some of Heidegger's critics.

It is these critics who stayed behind, while Jaspers and Heidegger set off together, alone.

Here alone does not mean by oneself but by themselves. See MacMillan:

used when two people are together and no one else is there

It was forbidden for an unmarried couple to be alone together.

Thus Heidegger and Jaspers depart together (in each other's company, not primarily: 'at the same time' although that is also true), alone (no one else is there, or by themselves).

This interpretation of the sentence would be clearer if the sentence had read

...The two then set off alone together.

Here, its is clear that the two departed alone, apart from the critics, but in each other's company (together).

The author flip flops alone and together and what this does is to stress that the two departed in each other's company and that the critics did not form part of this company. Thus the sentence is well written, the word order well chosen, and the meaning apparent if not upon first glance.