I was just wondering, if I have two variable x and y

x~N(0,1/2) and y~i*N(0,1/2)

Then would their sum x+y ~ N(0,1/2) + iN(0,1/2)

I don't think this is true but can someone show me if I'm right/wrong.

Also if I have x*i would this have distribution i*N(0,1/2)?