Can a tomato plant live indefinitely?

Like well before the frost kills it take a few suckers from it and root them inside in the window.  Then plant those rooted clippings in a small pots which you keep in the window over the winter.   Then transplant those in the spring outside.. rinse and repeat, over and over?

I would say, probably, yes.  Tomatoes, at least some varieties, can live multiple years in the right climate.  I think Anna Eddy had perennial tomatoes in her Solviva greenhouse on the New England Coast.

Short answer is yes.  There is an place about 300 or so miles north of me called Chena Hot Springs.  They have natural hot springs in the area and tap into that resource for many different things.  There is a resort of sorts there with heated pools that people use all winter, they heat all the buildings utilizing the free source of heat and actually generate all their own power from it as well.  All that long winddedness to say that they also have greenhouses that are heated and used year round and the last time I was there they had hydroponic tomato's that were about 50-75 feet long.  They are tied up with strings and as they grow the limbs are pruned off because the best production comes from the upper part of the plant and then they lower and wrap them around the growing area.  It was simply amazing to see.

https://chenahotsprings.com/chenagreenhouse  Direct link to the greenhouse section

https://chenahotsprings.com/renewables  Link to their renewables page

This is how named cultivars survive centuries they are rooted or grafted and just keep on going.
In the tropics tomatoes will last 8yrs or so, but indefinitely if grafted/rooted.

the latest vertical hydroponic growing of tomatoes has found they will produce continuously till they reach something like 36'