A DC / DC ATX typr PSU can be had relatively cheap.
Something along these lines.
Depending on some variables, like temperature a 12 volt SLA (Sealed Lead Acid) car/truck battery works out close to these numbers:
12.65 V = 100%
12.45 V = 75%
12.24 V = 50%
12.06 V = 25%
11.89 V or less = discharged
When fully charged about 2.1 volt/cell.
I believe people here were looking at this initially as an automotive application where battery voltages can vary between 12.6 to as high as 14.5 or greater system voltages.
You have a 12 volt monitor and a 12 volt battery and as Colin mentioned early on why do you want to regulate any voltages?
Do u think this could handle the high currents of a car battery or is there a better alternative?
The fact that the battery can source hundreds of amps means nothing. The load will only draw its rated current at its rated voltage. A 12 volt monitor rated at 4.5 amps likely will not draw that full current but a little less. Actually 4.5A X 12 Volts = 54 Watts which seems high to me for a LCD Monitor, even a 23 inch monitor. Mine typically draw about 23 to 30 watts.
Anyway, I would just run it as Colin suggested. If you really want 12 volts regulated from a 12 volt source then consider what I linked to. The only merit being most have automatic shutdown when the battery voltage drops below about 11.8 volts.
Just My Take
Ron