Telemachus is a complex character in The Odyssey by Homer. Explain.
In The Odyssey by Homer, Telemachus is, as you say, a very "complex character." Though he grows up in a rather privileged home, his circumstances certainly do not grant him many privileges. His father left when he was still a baby, so Telemachus grows up without a strong male presence, though he is certainly well loved by his mother.
He and his family have wealth, but the suitors which have descended on the household are consuming everything. Every day the suitors
come and loiter in our house and sacrifice our oxen and our sheep and our fat goats and make a holiday feast of it and drink the bright wine recklessly. Most of our substance is wasted.
Unfortunately, Telemachus begins to whine about his plight, wishing the "gods would give me such strength...to take revenge on the suitors for their overbearing oppression" but complaining that he and his mother simply have to take this abuse from the suitors. This reaction is not surprising, coming from a young man whose circumstances are so out of his own control.
He is a polite young man who knows the proper way to treat strangers, as he does with Athena when she arrives at the door in disguise.
[T]he heart within him [is] scandalized that a guest should still be standing at the doors.
When he sees the visitor has not been attended to, he immediately takes action. How awful and frustrating it must have been for a young man to know the proprieties of dealing with visitors but to be faced with the thankless beasts who arrived every day to wrangle his mother's hand in marriage. They were visitors and guests, but he had to be conflicted about the proprieties of how to treat these daily intruders.
Surely this was a difficult and complicated growing-up for a young boy, as he heard countless stories of his father's many moral qualities as well as his great strength but was not familiar with any of them personally. While Telemachus was undoubtedly proud of that heritage, he must also have been bitter about his father's leaving to fight for another family and not returning to fight on behalf of his own family. (We do know that Telemachus was scornful of his father's assumed drowning because that is such an ignoble death and brings no glory to Odysseus's family.) He probably wanted to do something to help him and his mother get out of this awful, unending predicament, but he lacks guidance and wisdom about how to go about it.
All of these unusual circumstances and complications begin to change when Athena (disguised as Mentor) starts to help Telemachus become a man of action (like his father) as well as a man of contemplation (unlike his father). When he returns from his journey, Telemachus is a qualified and capable partner for Odysseus, but that is not because of anything particular Odysseus did (which makes a case for genetics, I suppose).