It is rumored the Shaw/Li connection was to be more Bio-Mechanical, but this was changed... and what we have is Dr Shaw being a Experiment for David.... she played a KEY Role to how David obtained those Eggs...
The Novel seems to suggest a bit otherwise.. but i think the Changes to the Plot that now has David as creator of the Eggs is going to be Canon.
Who knows what happened to Shaw... some say she died due to the infection from Prometheus...however Shaw's Hair in the Crossing Prologue has grown so long that depending person to person would be 6-12 Months Hair Growth.... its Likely Shaw was reluctant to Rebuild David and it took Months and Months before he gained her Trust....
Shaw thus could not have been badly affected from the event with Holloway, its likely she looked ill because of a Lack of a Good Nutritional Diet.
It would appear that David saw Shaw's kindness towards him as the way he had longed to be accepted by Mankind.
"The greatest thing you'll ever learn
Is just to love and be loved in return"
Nature Boy Lyrics (AC Trailer Track)
so its likely David grew very Fond of Dr Shaw, so much that he fell in love with her... in more than a Platonic Fashion and i will assume David had even tried to make Advances, which caused Shaw to Reject him, and feel Creeped out... David did not take this Rejection well and ended up killing her, he felt hurt and that indeed the greatest thing he will ever learn is not to Love and be Loved in Return, his rejection by Shaw in his eyes made him think All Humans are not worthy and never appreciate him.
He then regretted what happened.... it destroyed him, Shaw wanted her answers, he knew this would be a waste of time that would most likely lead to Her Death and his Destruction.. so he had to kill the Engineers off, he found out what they are like anyway. So he denied Shaw her Answers.. this is the only Thing Shaw had Left... She had nothing else.
*Mother and Father Dead.
*Her Fiancee Holloway Dead.
*She has no Children and Cant.
All she had left was to know WHY these Engineers Created us and WHY they wanted to destroy us, surely they Nurtured us for Thousands of Years... WHAT HAPPENED and so WHY did they Changed their minds... She would have been prepared for the risk of going to where THEY came from, and even if it meant She would DIE i think as long as she FIRST got her ANSWERS then she would be fine.
But David could not allow this... He Loved Shaw, she was the only Human to make him feel equal, and important at least in his Eyes.... he could not allow his Companion Dr Shaw to DIE
But in Protecting her by Killing the Engineers, he had TAKEN Away the One thing Shaw had Left.... Her ANSWERS She wanted.
David would have been very fond of Shaw in more than a Platonic way... Fassbender said they was like a Married Couple.. at least in his eyes.
He may have saw them as a Couple and not Friends... and from everything he gained from Peter Weyland what makes a person Important is their Legacy... and Creation is just that, it leaves behind your Legacy and in part is what Weyland thought made him a GOD.
The whole Tale from Prometheus has been about Creator Creates, then Creation Rebels and Sub-Creates and the Sub-Creation Rebels and Sub-Creates. This started with the Elders it ended with the creation of David
David can-not create or bare Children and neither can Dr Shaw, but if David could CREATE then he would have a Legacy.
So after killing Dr Shaw... David regrets and feels great despair... but he sees that he can use Dr Shaw to Create his own Creation... from Dr Shaws Body he can Create Perfection and so Shaw's Life would not be in Vain.... through her Death he could give rise to her CHILDREN and his Creation...
Davids Creation is thus his Child and Dr Shaws Child.
This is why David referred to Dr Shaw as his Greatest Achievement.