The computer monitor gets its power from an adapter, while the adapter is connected to the a power strip with its own switch.
When the computer has been shut down, the monitor indicator keeps blinking.
Here, there are two things a user can perform.
Is the second step necessary? The reason a user will want to perform this is to have better longevity for the adapter.
With the exception of damage due to a severe power surge, such as that induced by a nearby lightning strike, there should be no detriment to the longevity of a modern LCD display with LED back-light if it is left on continuously. Older LCD displays with hot-cathode fluorescent back-light or cathode-ray tube (CRT) monitors might benefit from being turned off, but that's questionable: they have heated tungsten filaments. Leaving the filament warm causes it to evaporate (well, sublime) slightly, visibly darkening the glass near it, but cycling the filament on and off causes it to crystallize, crack and break.
Another reason to shut a monitor at the power switch, or to use a "smart" power strip to disconnect the mains is to conserve energy. However, a survey by Lawrence Berkeley National Laboratory shows the average standby (sleep, blinking indicator) power usage was about 1.4 W and in the "shutdown" (switched off) mode, the average consumption was about 1.1 W, hardly a significant difference. If your operating "off the grid", e.g. solar powered, then saving one watt might be helpful.
My own preference is to leave the LCD monitor switch in the on position unless there is a lightning storm in progress. BTW, I've found surge protectors valuable: when our mains neutral line came loose, they saved appliances, blowing fuses rather than burning out equipment (we did lose an unprotected clock and a fluorescent ballast, though).
