.bashrc

source ~/.rprompt.bash

export RPROMPT='$(date)'

.rprompt.bash

#!/bin/bash
#
# zsh style right prompt support :)


__rprompt() {
    printf "%*s\r" "$COLUMNS" "$(eval echo "$RPROMPT")"
}

if [[ ! $PROMPT_COMMAND =~ __rprompt ]]; then
    export PROMPT_COMMAND="__rprompt; ${PROMPT_COMMAND:-:}"
fi

If you want to customize the date formatting, do something like this:

export RPROMPT="\$(date '+%b %d %r %Z')"