In the solution, why is the the complex time not considered and real and positive time is considered? Is it because complex time does not exist or there is some other reason?
It's because complex time does not exist. There are many occasions in physics where a root is taken and a negative or complex solution is discarded because the quantity can only be positive (e.g. mass), or can only be real (time, or virtually any observable quantity in physics).
There is a use for the idea of imaginary time in relativity, but not in Newtonian Mechanics.
I believe a problem like this is dealt with in Paul Nahin's book The Story of (sqrt (-1)), An Imaginary Tale. If I understand this correctly, an accelerating pickpocket is trying to escape a jeep traveling at constant speed. The "physical" solution occurs in the unique case where the jeep is traveling just fast enough to catch up with and hence intersect the trajectory of the runner. The "unphysical" solution represents a situation in which the jeep is traveling fast enough to pass the pickpocket and keep going, after which the pickpocket catches up with the jeep and they meet once again at a second, later time.
Mathematical equations are models for the relationships among physical quantities in a physical situation. "Physically relevant solutions" are subject to implicit and explicit conditions placed on the various quantities. In some situations, one wants a time that occurs in the future, after the start of the problem---thus rejecting the negative quantity that might be a mathematical but not physically-relevant solution to the equations.
In your particular problem, as @RC_23 said, "complex time" doesn't exist... at least in typical physics problems. No clock-reading squared is negative. Similarly, no kinetic-energy is negative.
(If you did accept a complex time as a solution, what would that imply about other physical quantities that depended on that time? Would the resulting quantities be physically-relevant?)