R - compare two columns with NAs

I have this dataframe:

    df <- read.table(text="
                 date1       date2
    1            NA          2016-12-01 
    2            2017-01-01  2018-10-01 
    3            2016-12-01  NA 
    4            NA          NA
", header=TRUE)

What I need is to create new "max_date" column which will be containing maximum of date1/date2 columns. Note that at some rows are NA values, in some cases are NA in both columns.

I tried to achieve that with some if_else but code is too complicated.