Imaginary time concept in S.Hawking's No Boundary proposal, extra-time dimensions and the Big Bang In this post I will be refering to S.Hawking's lecture: http://www.hawking.org.uk/the-beginning-of-time.html
I have a couple of questions regarding the Imaginary time and the Big Bang.
In the article mentioned above, it is said that "The no boundary proposal, predicts that the universe would start at a single point, like the North Pole of the Earth. But this point wouldn't be a singularity, like the Big Bang".
This statement clearly tells us that the No Boundary condition and the Big Bang theory opposes each other with respect to initial singularity in real time.
My questions are:
1) Is it possible that the No Boundary condition proposal can still allow the Big Bang singularity in real time?
2)Can I just say that the concept of Imaginary time is equivalent to an extra-time dimension with certain properties and call it "t_2"? or How this Imaginary time dimension is different from extra time dimentions proposed in other theories like 3 time-dimensions theories?
I do not have mathematical background in Cosmology other than the standard level General Relativity, but these questions are free to be answered in any depth possible. Thank you!
Edited
3)How this No boundary condition explain the  origin of the universe without the Big Bang singularity when this singularity is intuitive?
 A: 1) Yes. This is a subtle aspect. At least in the paradigm advocated by Hawking, the geometry of the 4-manifolds included in the sum over histories is not to be identified with the classical Lorentzian histories that they provide probabilities for. The no boundary wave function resolves the initial singularity on the quantum level of the wave function. To get the classical history associated, you integrate back from the final surface using classical Einstein equations so singularities do occur. 
2) The slogan “rounding of the singularity in imaginary time" is misleading. In fact, there is no non-trivial example of a Euclidean instanton nucleating the universe. Roughly speaking, in order to make this integral converge one needs to include complex metrics, meaning no meaningful notion of signature! In order to satisfy the no boundary conditions, meaning a smooth cap off of the geometry, one generally can  and must take the metric in the initial evolution to approximately be Euclidean (so time complex) and approximately Lorentzian towards the end of the evolution. 
