Yip, although there's also length contraction in play. The universe just starts contorting itself to ensure nothing ever exceeds the speed of light.
What the GP post is referring to is called a 'relativistic rocket'. You can find plenty of calculators for them, like here [1]. It leads to really mind-bending scenarios. For instance a single human could easily travel a billion light years within their lifetime, requiring "only" to travel at 1G acceleration for 40 years. Of course in that time a billion years would have really passed outside of our relativistic rocket.
If it ultimately turns out that there is no fundamental law of the universe that makes long-term rates of relatively low acceleration impossible, then the future will be a simply unimaginable place. Time itself will start to lose meaning as it becomes as variable as distance is today, at least for those able to travel.
[1] - https://www.anycalculator.com/starshipcalculator.html