This is precisely the question which I tasked myself with several years ago. I've been developing games for more years than I can remember, and am currently doing so for Deen Games. So this is a question that falls within my interests.
In short, Islam allows all fields of interest, unless they contain haram elements. If they do contain haram elements, those elements can be removed.
Regarding pictures, the prohibition of tasweer or picture-making does not apply to digital artwork. This is the majority opinion and the correct opinion: tasweer is only when you physically draw or print pictures.
The proof of this is complicated. It has to do with the fact that photography didn't exist at the time of the Prophet (peace be upon him). Some scholars made qiyaas that picture-taking is like drawing, although the majority do not, and offer then that digital artwork is nothing like physical drawing; therefore, all the rulings of tasweer don't apply to it.
For a proof, you can read this fatwa on Islam-QA. I quote (emphasis mine):
With regard to pictures made in the modern fashion, they fall into two
The first category is those which have no tangible substance (and can
only be seen by running them through a machine), as I was told is the
case with pictures on video tapes. There is no ruling at all
concerning these, and they do not come under the prohibition at all.
Hence the scholars who forbid making pictures with cameras on paper
(photographs) permitted this (video pictures), and said that there is
nothing wrong with this. [...]
"but game development always does not include living pictures". This is what i am asking. If it doesn't include pictures of living beings, is it permissible then?. Many games involve only cartoon characters.