I wonder how long it'd take, seen from the perspective of the ship, if we had an engine capable of delivering a constant 1g acceleration.
That's the way we'll want to travel, after all.
The faster you go, the higher your relativistic mass, so if you accelerate by constant force, the lower your acceleration. Conversely, to keep acceleration constant, you would need larger and larger forces, in the limit ‘infinite force’.
For more and better info, see https://en.wikipedia.org/wiki/Space_travel_using_constant_ac....
The Wikipedia article you linked to calls this (rightly) a "half-myth". It depends on your reference frame.
The spaceship's own reference frame will be a whole different story. If I understand correctly, it will perceive its own mass to be constant, the rest of the universe's relative speed to asymptotically approach the speed of light, but also the rest of the universe to get progressively squashed in the direction of travel (length contraction dual to time dilation), with the two effects multiplying up in such a fashion that the Newtonian relation between the integral of your acceleration and the time it takes to the destination seems to hold throughout.
In other words, for the spacefarer's schedule, it's as if the speed of light does not actually play a role: they can always accelerate more/go arbitrarily faster to get to their destination faster. However, from their point of view, it is as if the universe around them will deform in the process to accommodate this, and from the stationary observer's point of view, it is as if they are actually still moving slowly but tricking themselves into thinking otherwise by accelerating their passage of time.
Seen from Earth, it'll be around 13 years (takes a year to got close to c and another to decelerate - that is two years to travel roughly a lightyear, then 11 to travel 11).