If someone wears a garment with four corners, it needs tzitzit. Now, a cape pretty clearly has four corners. However, it's not clear to me whether or not "wearing" it follows the halachic definition of wearing. A cape is (generally) worn by tying two of the corners around your neck, and letting the rest hang behind you. You know, Superman style. In this manner, it doesn't provide any protection or cover you up at all. In fact, the only thing it seems to be good for is for looking cool while flying through the air.
EDIT: My initial reading of the shulchan aruch was wrong. According to the cited sources, a cape such as described in the question (a typical superhero cape), would not require tzitzit as it only has two corners at the bottom
According to the shulchan aruch (O.C. 10:7 and 10:8) a cape as you have described (open up front, and usualy tied up top) would require tzitzit:
Shulchan Arukh, Orach Chayyim 10:7:
מלבושים שהם פתוחים מן הצדדין למטה ויש להם ד' כנפות לצד מטה ולמעלה הם סתומים אם רובו סתום פטור ואם רובו פתוח חייב. ואם חציו סתום וחציו פתוח מטילין אותו לחומרא וחייב בציצית ואין יוצאין בו בשבת:
[Regarding] Garments which are open from the sides down and have four corners on the lower part but on the upper part they are closed: if the majority is closed they are exempt [from tzitzit] while if the majority is open they require tzitzit. And if half is open and half is closed one treats it stringently and it requires tzitzit, but one must not go out [in public domain] wearing it on the Sabbath.
and Shulchan Arukh, Orach Chayyim 10:8:
קאפ"ה שהיא פתוחה בענין שיש לה ד' כנפות אם יקבעו בה אשטרינג"ה לעשותה כסתומה כדי לפוטרה מציצית אינו מועיל תקון זה אם לא תהיה קבועה מחצי ארכה ולמטה לכל הפחות וגם שתהיה קבועה למטה מהחגור למען יהיה הרוב הסתום רוב הנראה לעינים דאם לא כן יאסר משום מראית העין:
A kapa which is open, in a way in which it has four corners: if one has affixed astringa to close it up in order to exempt it from tzitzit, it is ineffective unless it fastens for at least half the length and down to the bottom and should also be fastened below the belt in order that the majority should be closed, a majority that appears to the eyes; if not, it is forbidden because of marit ayin
The primary distinction seems to be whether the garment is closed up front (generally called a cloak) or not majorly closed up front (generally called a cape). The latter fits into the description in your question.