Originally Posted by Jeff Root

No, not in general.

If you were carrying all your propellant, you would need to
reduce your thrust as the mass of propellant decreased.

If your propellant magically appeared in your fuel tank just
at the moment you needed it, moving at the same speed as your
ship, you would not need to change your thrust at all to
maintain constant acceleration forever.

If you were propelled by light pressure from a laser beam
back in the solar system you are leaving, the power of the
laser beam would need to increase as your speed increased,
in order to maintain constant acceleration-- even if the
beam didn't spread out beyond the edges of your lightsail
at great distances.

Relativistic effects don't affect how you perceive yourself,
only how you perceive others, and how others perceive you.
It doesn't matter how fast you are moving relative to someone
else, you can always accelerate at 1 g with the same thrust,
in any direction. But if you are moving at nearly c relative
to me, I will perceive that your 1 g forward acceleration
doesn't increase your speed relative to me by much. To you,
the trip will be made shorter because the Universe appears
to shrink significantly in the direction you are traveling.

-- Jeff, in Minneapolis