One main issue is that income inequality decreases socioeconomic mobility by inflating the cost of scarce goods that improve earnings capabilities. So 4K TVs are cheaper but college tuition becomes unaffordable, because good universities and grad programs are valuable, scarce goods that the wealthy can outbid everyone else on.
Policies that aggressively expand capacity and defray costs, such as post-WW2 expansions of universities and the GI bill, can ameliorate this by shifting monies down the socioeconomic ladder via taxation, but high levels of inequality also lead to a powerful political interest group that generally seeks to reduce tax rates that affect them.
There's also the more general economic argument that high levels of inequality are economically inefficient since they result in a lower average marginal propensity to consume, but that's not so much "bad" as "could be better and provide greater macro stability."