I'm running todoist on chrom. When I set a task for today (or any date), it assigns the task to the next day.
Please make sure that you're using the same timezone in Todoist as on all devices you're accessing Todoist on and in case it asks you if you want to adjust the time zone, please accept.
If the issue persists, please tell me - when you set a task to a date with time such as "today at 16:00", does it change the time too? And if so - how much?