How to calculate current from a phasor?
Seemingly you know that your expression presents a rotating vector in the complex plane. As technical description of a current it's incomplete. The expression is without dimensions (=amperes, seconds, angle units). The variable Io of course can have any dimension because it's a multiplier. But if the expression presents current, then Io should finally get numerical values only as amperes or other units for current.

It's very common to leave all dimensions out and agree "SI units are used everywhere and the angles are dimensionless radians"" Assuming that the phase shift Pi/3 and the period 1/60 seconds (that's 60 Hz mains frequency) are ok.

The actual current is sinusoidal. It can as well be the real part of your vector or the imaginary part of your vector.

Texts which handle the theory behind the practical mains power electricity generally present currents and voltages with sine functions and there your expression presents current which as a function of time is the imaginary part of your expression (without the j). Deriving theoretical formulas use Io as the peak current. But practical numeric calculations are done with RMS length phasors and of course omitting the rotation, having only the phase shifts between different currents and voltages shown in complex phasors.

Theoretical signal mathematics and communication theory texts have their own conventions. Formulas are derived with the complex expressions but actual current or voltage is only the real part. Instead of the real part also the actual signal in theoretical texts can be complex like yours. 

If one really wants to have a physical realization with analog circuitry, real and imaginary parts are both in different wires. As computer data the real and imaginary parts are stored like any numbers. Analog physical realizations with real and imaginary parts in separate wires actually has been used in advanced modulator, detector and filter circuits for ex. in radars.

Finally there exists 2 phase AC power systems where the phase difference is 90 degrees. There your expression can present a real current as is. One wire is for the real part and the other wire is for the imaginary part. The third wire is the neutral common. Systems like this are not common. I have seen only single motors+ their supplying circuits with it, never even a house wide power distribution system. Electromechanical control systems have use it because precise rotation angle measuring was and still is possible with 2 phase AC resolvers.