I think Jeff and Faultline are both right, it's a question of reference frame. If you could maintain a 1G acceleration in your own frame, it wouldn't require any changes to the engineering of your ship, you'd just have to do it without running out of fuel (a major challenge!). From the perspective of the people back on Earth, you would not be accelerating this fast once you became relativistic, they would see all your increase of momentum going into increased relativistic mass rather than velocity.

The frame of the rocket is weird, because it is an accelerating reference frame. A 1 g acceleration would mean that objects moving with you would fall back by 9.8 m/s for every second that you accelerate. Bute note, if you notice alpha Centauri approaching you at c minus 9.8 m/s at some point in this process, if you would not see it approaching at c a second later. The 9.8 m/s would only apply to objects at rest with respect to you, the velocity addition formula would still have alpha Cen approaching at just a little faster than c minus 9.8 m/s. So it would be hard to accelerate the approach of alpha Cen as you become relativistic. No matter, what you would instead see is a shortening of the distance to alpha Cen, not quite the same thing as alpha Cen moving toward you! Back on Earth, people would not see alpha Cen approaching you at faster than the speed of light, but they would see time slowing down for you. At the end of the day, you could go very far in a short amount of time if you could really maintain 1 g. The energy challenges are another matter, I'll have to think about that, but it's clear that you'd have to convert most of the rest mass of your ship into energy to make it work.