I have bought a Chinese Elm last year and put it outside on the balcony since then. It seems to be doing well and it is now growing new leaves.
Would it be a problem to bring my bonsai inside during the growing season? I find it more aesthetically pleasing to watch my bonsai on the window sill.
When winter is coming I would put it outside again for their natural dormancy cycle.
Yes. It would be a problem. Inside it won't (and cant) prepare itself for winter dormancy. Dormancy is not simply putting the tree out in the cold. Dormancy is a process that temperate zone plants begin in June. Shortening daylength(beginning at the Summer Solstice-when days start getting shorter) and gradually declining temps from late summer into fall, along with frost and shallow freezes in mid-late fall harden off tree roots so they are prepared to withstand deeper winter cold.
Your tree will miss all of that inside, as lighting on a windowsill is nowhere near the natural light exposure your tree needs. Temperatures inside (aside from the extremely low desert-like humidity and cave like lighting levels) will remain constant so your tree will continue growing well into the time it should be preparing for winter.
If you keep the tree inside all summer, then plunk it outside in the cold, there is a chance you will shock it into dropping leaves-- which isn't dormancy...and given the right circumstances outside and the tree's weakened condition (Compared to elms kept outside all year, it will be weak from lack of light and stressful indoor conditions overall), you could kill it.
The bottom line here is that you may prefer having the tree on a windowsill indoors, but the tree doesn't care about you aesthetic choices. It cares about living. It's needs--like it or not--are the priority. The hard truth is that failure to meet those needs results in a dead tree.