I'm starting to get suspicious about this car. In another thread that I've only just posted in, your car's 4WD warning light comes on when water is splashed underneath the car. Now when you accelerate hard, your alternator and handbrake lights come on.
The only relationships between these things are:
* They're in the same car. Had to be said.
* Same battery providing the power. Don't bother, it won't be this.
* The wiring loom to the instrument panel (good chance it's this).
The handbrake light is more than likely managed by a signal from the BCM, the alternator by a signal from the ECU, the 4WD light by the transfer case. But they ALL send a signal to the instrument cluster via the one loom ...
So I'd look under the dash and see if that loom has been crushed by someone installing a stereo or an alarm or something. Maybe the loom has chafed against the steering column, or brake pedal or something. There's just no logical reason for these three things to come on like you've described unless there's some loom damage, and of course then anything's possible.
Has the car been off-road much? Has anyone installed anything inside?