1.After completing his master's degree in music, john decided to pursue doctoral study. 我之前打少了一句?

1. He returned to his country having completed his studies.
2. He returned to his country after (he) completed his studies.
以上兩句來自Yahoo 英文知識網站，經調整後
老外認定無差異，老外原始意見如下
The first one looks fine to me. The second one seems to be missing a subject pronoun: He returned to his country after he completed his studies. I wouldn't use "after completed his studies" by itself. Using "after his studies were completed" would also be fine.
3. Can a person change his name after completing his
graduation?
這句也來自Yahoo 英文知識網站議題
4. After completing his research abroad, Dr. Ko returned to
Taiwan to lead an organ transplant team targeted at
improving the success rates of heart transplants.
這一句大家看的出，人物是指柯文則
5. Having completed his sentence for the unintentional crime that derailed his youthful plans for fame and fortune, Levi Grant looks to start over in the town of Spencer, Texas.
6. Having completed his programme in Tanzania, President
Erdoğan proceeded to Ma
7.After completing his master’s degree, he took up his training in allergy at the world-famous Karolinska Hospital in Sweden. In 1990, he added another feather to his cap, when he was sent on another government scholarship to Mayo Clinic (USA) to pursue his ...
8.London Arts and Humanities Partnership (LAHP) The School is a member of the AHRC-funded consortium LAHP, along with King’s College London and UCL. Up to 80 studentships are available for postgraduate research students wishing to pursue doctoral study in ...
再参考以上四句無瑕疵句， 實在看不出所列二句有何不同或有任何文法瑕疵
且所有用詞均屬正確並合理

不應該是After obtaining his master's degree in music, John decided to go for a doctorate.
Having obtained his master's degree in music, John decided to go for a doctorate.

應該寫成：
After obtaining his master's degree in music, John decided to go for a doctorate.
Having obtained his master's degree in music, John decided to go for a doctorate.
前者的修飾語是一個「介系詞片語」（介系詞after＋動名詞片語obtaining ...）
後者的修飾語是一個「分詞構句」（用完成式分詞having obtained ...，表示其時間在主要動詞decided之前）
兩個修飾語都是述部(decided ....)的修飾語，屬於副詞性質。
這兩句只有文法結構上的不同，在意義上並無不同。

你的原句用complete可能用問題，因為學位應用取得(obtain)，不是完成(complete)；此外，study也不會與pursue這個動詞搭配。