Is a bachelor’s degree in computer science enough for a career in software engineering?
An undergraduate degree should prepare you for the job market - for many (if not most) jobs.
As a software engineer you will evolve over time learning new programming languages and platforms - essentially the tricks of the trade.
A masters degree will give more of that training upfront with a more solid background that should serve you for many years to come.
Having said it, it is not necessary to get an masters degree to become a great software engineer. Also, since masters degrees are typically a specialization in one field, unless you know what you want to specialize in, you might be better off gaining a masters somewhere down the road especially if you plan on moving to a different country (with a slightly different job market).