Why is it "going off this morning" instead of "went off"?

Going off is not present continuous. It is not any tense, because it is not a finite verb.

It is a participial clause, modifying the noun phrase your alarm clock, which is the object of hear. It would not be grammatical to use the finite clause went off in that context.

The person making the statement is describing something which happened in the past. However, the situation being described continued for a while.

Let's say at 9am the speaker is referencing the alarm which started to ring at 7am. The speaker doesn't just mean that at 7am the alarm rang one time.

The speaker means that at 7am it started to ring, and continued to ring, and the owner of the alarm continued to let it ring.

That's the implicit question: "Didn't you hear it ringing? How come you let it ring and didn't get out of bed/didn't turn it off/etc."

It's implying that during the duration of time when the alarm was going off, there should have been some action (which didn't occur).

So it's not just "in the past something happened" but "in the past there was a situation during which something should have happened."