Serial killer in a dream: what does it mean?
The figure appears in dreams to people with no connection to any such thing, and it is a media figure in dreams far more often than a personal one.
How do the four traditions read it?
Each reading below comes from a named tradition and a cited source, so you can see where it came from and decide what fits your dream. The four often disagree, and that is the point: a symbol rarely means one fixed thing, and the distance between Freud, Jung, modern research and myth tells you more than any single verdict could.
Freud
Freud would treat a pursuing figure as something the dreamer has put outside themselves, and would ask what has been given to a stranger to carry.
After Freud, The Interpretation of Dreams (Brill trans., 1913), public domain.Jung
A Jungian reading would call this shadow material and would be careful about it, since the point of the concept is that it is recognised rather than acted on, and a dream is exactly where recognition can happen safely.
Our summary of a Jungian reading, in our own words, not Jung's. See Jung, ed., Man and His Symbols (1964).Modern
Being chased is among the most commonly reported dreams anywhere, and the identity of the pursuer is usually supplied by whatever the dreamer has been watching. There is no association between violent dream content and violent behaviour, and dreaming this says nothing about anyone.
Our summary of the typical-dreams literature. See Nielsen et al., Dreaming 13(4), 2003, 211-235.Mythological
The figure as a category is recent, and the older equivalents are monsters and demons rather than people, which is a meaningful difference in how a threat was understood.
Rosmerta Publishing editorial synthesis (after Frazer).How can writers use this symbol?
The same symbol, read as craft material rather than as a dream. What it does on the page, how genres use it differently, and what happens when you turn it inside out. Written for novelists, screenwriters and poets who need the symbol to actually work, and it ends with something to try.
In literature
The genre has an ethical problem it mostly declines to notice, since it takes real crimes against real people and converts them into a puzzle with an interesting antagonist.
Rosmerta Publishing editorial synthesis. Our own words; no quoted text.What it does in a story
It supplies a threat with no motive a reader can share, which removes negotiation as an option.
Genre conventions
Crime fiction gives them an elaborate method. The documented reality is squalid and repetitive, which almost nobody writes.
How to subvert it
Write the investigators, exhausted, doing paperwork, getting nowhere for eleven months.
Try this
Write four hundred words of a case review meeting.
Only the first of these names a source. What a symbol does in a story, how genres treat it, how to invert it and what to try are our editorial judgement as writers and editors, not findings from a body of research. They are offered as craft opinion and should be argued with.