Plenty of plug in dimmers do the same but it does not work well if it all with modern bulbs.
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I don't think you'll find one these days. The problem is the trickle current is enough to make some LED's glow dimly all the time since they use so little current. I use an Insteon system in my house and when I used an older dimming lighting module with that feature to try to run my LED Christmas lights they would always stay dimly lit even when switched off.
You have a good memory. I know that several Z-Wave dimmer lamp modules (e.g. Linear/GoControl/2Gig PD300Z) have this feature. Some manufacturers call it Auto-sense, but it might have other names at other vendors.
Could you expand a little on the situation / problem you want to solve?
I concur with the others that the exact thing as asked may be a bit tricky, but there may be different solutions to your underlying problem, that don't work via flicking the lamp switch.
I'm asking this, because I find myself in a similar, but different situation. Being less about plugged-in lamps and more about my ceiling lights that I'd like to "smartify". But I can't really put smart relays in the ceiling, as they'd be rather quite difficult to access if they need maintenance. The solution for that is to install smartified wall switches, that can both act on an automation command as well as on using the physical light switch.
For ceiling lights wiring that I can access for maintenance, there exist small modules that can be put in line and act both on a normal light switch as well as a "smart" signal.
Of course none of this would be a solution for a desk lamp or uplighter. But getting an independent wireless button to use instead of the switch in the lamp cord could provide local control, at the cost of introducing an extra device instead of using the existing switch.
That kind of thinking is what leads me to ask up there: What's the underlying problem that the smart plug is intended to solve?