What is happening neurochemically when we seek to reduce stress by taking aggressive action? As a layman, it seems to me that humans have 2 primary strategies for controlling stress (cortisol). I tend to use the words “connection” and “aggression” to refer to these strategies.

The “connection strategy” involves deepening an empathetic connection with others.
This deeper connection enables us to be aware of their current emotional state and thus to set them at ease which opens them up to be influenced.
Because we can now feel their emotional state in real time, we can calibrate our interactions with them to influence their thoughts, feelings and actions more skillfully.

I like to think of this as the Non Violent Communication strategy used by Gandhi, Martin Luther King, Nelson Mandela, etc. to remain calm even when threatened with danger.

It seems that oxytocin is involved in this connection strategy.
When our response to stress involves shifting our focus and intention to connection, oxytocin increases and cortisol decreases. (I think)

The “aggression strategy”, on the other hand, seems to involve involves taking determined action to conquer something that we believe to be causing the stress.

I am guessing that hormones like adrenaline and maybe testosterone are involved in this process but I’m not sure.

What is actually happening neurochemically when we get aggressive?

I believe that serotonin is involved in both strategies.
Serotonin being the neurochemical that is released when we have confidence that our needs can be met.
I believe that both the connection and aggression strategies seek to reduce cortisol and increase serotonin, but I would really like to understand exactly how this works.
Especially for the case of aggression.
So there's a lot of things to talk about here.

First off, there are many strategies one can use to regulate stress.  James Gross, for example, has put forward an incredibly influential process model of the strategies we might use. His work and others have pointed to:


Distraction
Reappraisal (reinterpreting the situation)
Expressive suppression
Thought suppression (Wegner)
Rumination (Nolen-Hoeksema)
Worry
Self-distancing (Ayduk, Kross)
Situation selection
Situation modification
Interpersonal regulation (social support)
Acceptance (Linehan)
Exercise
Social sharing (Rimé)
Externalizing behavior (e.g., aggression)
Drugs
And more…


Not all of these are "effective," but you can see that there are a lot of ways to cope with stress.

When talking about aggression, there's likely a bunch of processes going on that could change depending on the context and individual.  The psychophysiology and neurochemical changes from one instance to another, and from one person to another, are likely not going to be the same. So there's not really one answer to your question.

Indeed, aggression:


can be negatively or positively valenced (i.e., it can feel good or bad; e.g., Chester &amp; DeWall, 2015)
might involve some sort of external or internal focus (recruiting relatively more exteroceptive attentional networks, e.g., Wager et al., 2015 or interoceptive cortex, e.g., Dambacher et al., 2015, respectively)
might involve a lot of top-down regulation (e.g., Achterberg et al., 2016) or be largely impulsive
might motivate approach or withdrawal behaviors


And this logic extends to neurotransmitters, neuropeptides, and hormones.  Their activity depends on the context and the person (over time).  In general, you can probably expect a complex interaction among these things, and involvement of practically all of them.

So as with most things in psychology, the answer is: it depends!