In general, court hearings are held in public; and anyone is free to make a fair and balanced report of the proceedings. It is important that justice is seen to be done.
However, in some circumstances that is not true—the obvious example is if a child is involved. It is not sufficient though merely that "[the claimant] wants to keep his/her identity confidential".
Have a look at the Civil Procedure Rules (part 39):
39.2 General rule – hearing to be in public
The general rule is that a hearing [including a trial] is to be in public.
A hearing, or any part of it, may be in private if –
(a) publicity would defeat the object of the hearing;
(b) it involves matters relating to national security;
(c) it involves confidential information (including information relating to personal financial matters) and publicity would damage that confidentiality;
(d) a private hearing is necessary to protect the interests of any child or protected party;
(e) it is a hearing of an application made without notice and it would be unjust to any respondent for there to be a public hearing;
(f) it involves uncontentious matters arising in the administration of trusts or in the administration of a deceased person’s estate; or
(g) the court considers this to be necessary, in the interests of justice.
The court may order that the identity of any party or witness must not be disclosed if it considers non-disclosure necessary in order to protect the interests of that party or witness.
If you can satisfy the court that one of (a) to (g) apply, the court may make an order anonymizing proceedings (for instance, you would only be referred to as "AB" in reports and judgements) or it might hold the hearing in camera (in private).
However the other party may well wish to oppose your application. Indeed, even a newspaper hearing of your case but unable to report it is entitled to object: http://www.bailii.org/uk/cases/UKSC/2010/1.html (In re Guardian News)
In these circumstances, when carrying out the ultimate test of balancing all the factors relating to both M's article 8 rights and the article 10 rights of the press, we have come to the conclusion that there is indeed a powerful general, public interest in identifying M in any report of these important proceedings which justifies curtailment, to that extent, of his, and his family's, article 8 Convention rights to respect for their private and family life.
For all these reasons, we would set aside the anonymity order in respect of M. ... Therefore, A, K, M and HAY will be named here and in the judgments on the substantive appeals, as Mr Mohammed Jabar Ahmed, Mr Mohammed Azmir Khan, Mr Michael Marteen (formerly known as Mohammed Tunveer Ahmed) and Mr Hani El Sayed Sabaei Youssef (or Hani al-Seba'i), respectively.