CT constantly trims your plane for level flight at full throttle regardless of what you are currently doing. Its only input is your current IAS.
In a dog fight the problem with CT is that it tends to make me bleed too much energy in a turn - If you hold the stick at a constant deflection, as your plane loses speed in the turn it keeps triming you nose-up, i.e. increase your angle of attack for the stick deflection that you are holding. For this reason I tend to keep it off most of the time.
In a dive, CT will keep triming you nose-down as speed builds up, making pull out ever more difficult. On planes like the 109, P-38, mosquito and some others it is almost a suicide device.
On the other hand, you generally do not want to be far off the "neutral" (i.e. level flight) trim. The compromise I adopted is to use the CT as a quick rough trimmer. When trim starts to bother me and I do not want to mess with the manual trims I turn it on for a second and immediately off again. I have it mapped to a button on my stick and go click-click when my speed changes and I feel "out of trim". In some planes the rudder needs a lot of trim input as speed varies (F6F for example, mosquito to a lesser degree). It is important to keep the slip/skid ball centered to conserved energy. Click-click, problem solved. I pulled some hard break turn and lost lots of speed, I find I keep constant back pressure on the stick - click-click, done. I am as close to trimmed as I need. For non-maneuvering flight, you have the auto-level, auto-angle and auto-speed (aka auto climb) trims.
One warning about turning CT on & off - do not do it while you are pulling Gs and in particular when the stall buzzer is buzzing. It will "jump" your trim instantaneously and you risk it pushing you beyond the stall and snapping the plane into a spin.