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.
1Sorry, but your reading if the SA is incorrect. It says specifically "and has four corners on the lower part but closed on the upper... " In this case, the cape has TWO corners on the lower part, and two tired together (closed) on the upper. No tzitzit here. May 20, 2015 at 11:10
Oops, you seem to be correct. Apparantly my lack of experiences with actual capes made me not take notice of this partical detail in the text. Tbh, even though I accept your correction, I have difficulty imagining such a cape.– RonPMay 20, 2015 at 11:47
2you never tied a blanket around your neck and pretended to be super man when you were younger? May 20, 2015 at 12:12
I still do that from time to time ;P. But indeed, most, if not all classic superhero capes would not need tzitzit, since the bottom only has two corners. Given that the kapa referred to here is ladino for a large type of cape/cloak similar to an overcoat, I guess that the four-corner types of kapa would be capes that have a deep split in the back? also: do I need to add an edit to the answer to make clear that my first interpretation was in error?– RonPMay 20, 2015 at 14:20